Integral Methods in Science and Engineering, Volume 1 Theoretical Techniques

Christian Constanda Matteo Dalla Riva Pier Domenico Lamberti Paolo Musolino Editors Integral Methods in Science and Engineering, Volume 1 Theoretical Techniques

Christian Constanda • Matteo Dalla Riva Pier Domenico Lamberti • Paolo Musolino Editors

Integral Methods in Science and Engineering, Volume 1 Theoretical Techniques Editors Christian Constanda Matteo Dalla Riva Department of Mathematics Department of Mathematics The University of Tulsa The University of Tulsa Tulsa, OK, USA Tulsa, OK, USA

Pier Domenico Lamberti Paolo Musolino Department of Mathematics Systems Analysis, Prognosis and Control University of Padova Fraunhofer Institute for Industrial Math Padova, Italy Kaiserslautern, Germany

ISBN 978-3-319-59383-8 ISBN 978-3-319-59384-5 (eBook) DOI 10.1007/978-3-319-59384-5

Library of Congress Control Number: 2017948080

© Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface

The international conferences on Integral Methods in Science and Engineering (IMSE), started in 1985, are attended by researchers in all types of theoretical and applied fields, whose output is characterized by the use of a wide variety of integration techniques. Such methods are very important to practitioners as they boast, among other advantages, a high degree of efficiency, elegance, and generality. The first 13 IMSE conferences took place in venues all over the world: 1985, 1990: University of Texas at Arlington, USA 1993: Tohoku University, Sendai, Japan 1996: University of Oulu, Finland 1998: Michigan Technological University, Houghton, MI, USA 2000: Banff, AB, Canada (organized by the University of Alberta, Edmonton) 2002: University of Saint-Étienne, France 2004: University of Central Florida, Orlando, FL, USA 2006: Niagara Falls, ON, Canada (organized by the University of Waterloo) 2008: University of Cantabria, Santander, Spain 2010: University of Brighton, UK 2012: Bento Gonçalves, Brazil (organized by the Federal University of Rio Grande do Sul) 2014: Karlsruhe Institute of Technology, Germany The 2016 event, the fourteenth in the series, was hosted by the University of Padova, Italy, July 25–29, and gathered participants from 26 countries on five continents, enhancing the recognition of the IMSE conferences as an established international forum where scientists and engineers have the opportunity to interact in a direct exchange of promising novel ideas and cutting-edge methodologies. The Organizing Committee of the conference was comprised of Massimo Lanza de Cristoforis (University of Padova), chairman, Matteo Dalla Riva (The University of Tulsa) Mirela Kohr (Babes–Bolyai University of Cluj–Napoca), Pier Domenico Lamberti (University of Padova),

v vi Preface

Flavia Lanzara (La Sapienza University of Rome), and Paolo Musolino (Aberystwyth University), assisted by Davide Buoso, Gaspare Da Fies, Francesco Ferraresso, Paolo Luzzini, Riccardo Molinarolo, Luigi Provenzano, and Roman Pukhtaievych. IMSE 2016 maintained the tradition of high standards set at the previous meetings in the series, which was made possible by the partial financial support received from the following: The International Union of Pure and Applied Physics (IUPAP) GruppoNazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), INDAM The International Society for Analysis, Its Applications and Computation (ISAAC); The Department of Mathematics, University of Padova The participants and the Organizing Committee wish to thank all these agencies for their contribution to the unqualified success of the conference. IMSE 2016 included four minisymposia: Asymptotic Analysis: Homogenization and Thin Structures; organizer: M.E. Pérez (University of Cantabria) Mathematical Modeling of Bridges; organizers: E. Berchio (Polytechnic University of Torino) and A. Ferrero (University of Eastern Piedmont) Wave Phenomena; organizer: W. Dörfler (Karlsruhe Institute of Technology) Wiener-Hopf Techniques and Their Applications; organizers: G. Mishuris (Aberys- twyth University), S. Rogosin (University of Belarus), and M. Dubatovskaya (University of Belarus) The next IMSE conference will be held at the University of Brighton, UK, in July 2018. Further details will be posted in due course on the conference web site blogs. brighton.ac.uk/imse2018. The peer-reviewed chapters of these two volumes, arranged alphabetically by first author’s name, are based on 58 papers from among those presented in Padova. The editors would like to thank the reviewers for their valuable help and the staff at Birkhäuser-New York for their courteous and professional handling of the publication process.

Tulsa, OK, USA Christian Constanda March 2017 Preface vii

The International Steering Committee of IMSE: Christian Constanda (The University of Tulsa), chairman Bardo E.J. Bodmann (Federal University of Rio Grande do Sul) Haroldo F. de Campos Velho (INPE, Saõ José dos Campos) Paul J. Harris (University of Brighton) Andreas Kirsch (Karlsruhe Institute of Technology) Mirela Kohr (Babes-Bolyai University of Cluj-Napoca) Massimo Lanza de Cristoforis (University of Padova) Sergey Mikhailov (Brunel University of West London) Dorina Mitrea (University of Missouri-Columbia) Marius Mitrea (University of Missouri-Columbia) David Natroshvili (Georgian Technical University) Maria Eugenia Pérez (University of Cantabria) Ovadia Shoham (The University of Tulsa) Iain W. Stewart (University of Dundee) A novel feature at IMSE 2016 was an exhibition of digital art that consisted of seven portraits of participants and a special conference poster, executed by artist Walid Ben Medjedel using eight different techniques. The exhibition generated considerable interest among the participants, as it illustrated the subtle connection between digital art and mathematics. The portraits, in alphabetical order by subject, and the poster have been reduced to scale and reproduced on the next two pages. viii Preface

Digital Art by Walid Ben Medjedel Preface ix Contents

1AnL1-Product-Integration Method in Astrophysics...... 1 M. Ahues Blanchait and H. Kaboul 1.1 Introduction...... 1 1.2 A Product-Integration Method in L1.Œa; b; ¼/ ...... 2 1.3 Iterative Reﬁnement ...... 4 1.4 Numerical Evidence...... 5 References...... 7 2 Differential Operators and Approximation Processes Generated by Markov Operators ...... 9 F. Altomare, M. Cappelletti Montano, V. Leonessa, and I. Ra¸sa 2.1 Introduction...... 9 2.2 Canonical Elliptic Second-Order Differential Operators and Bernstein-Schnabl Operators...... 10 2.3 Other Classes of Differential Operators and Approximation Processes...... 16 2.4 Final Remarks ...... 17 References...... 18 3 Analysis of Boundary-Domain Integral Equations for Variable-Coefﬁcient Neumann BVP in 2D...... 21 T.G. Ayele, T.T. Dufera, and S.E. Mikhailov 3.1 Preliminaries...... 21 3.2 Parametrix-Based Potential Operators ...... 23 3.3 BDIEs for Neumann BVP ...... 26 3.4 Equivalence and Invertibility Theorems...... 27 3.5 Perturbed BDIE Systems for the Neumann Problem ...... 30 3.6 Conclusion ...... 32 References...... 33

xi xii Contents

4 A Measure of the Torsional Performances of Partially Hinged Rectangular Plates...... 35 E. Berchio, D. Buoso, and F. Gazzola 4.1 Introduction...... 35 4.2 Variational Setting and Gap Function Definition ...... 36 4.3 Proof of Theorem 1 ...... 40 4.4 Proof of Theorem 2 ...... 41 4.5 Proofs of Theorems 3 and 4...... 43 References...... 45 5 On a Class of Integral Equations Involving Kernels of Cosine and Sine Type ...... 47 L.P. Castro, R.C. Guerra, and N.M. Tuan 5.1 Introduction...... 47 5.2 Integral Equations Generated by an Integral Operator with Cosine and Sine Kernels ...... 48 5.3 Operator Properties...... 53 5.3.1 Invertibility and Spectrum...... 53 5.3.2 Parseval-Type Identity and Unitary Properties ...... 54 5.3.3 Involution ...... 55 5.3.4 New Convolution ...... 55 References...... 56 6 The Simple-Layer Potential Approach to the Dirichlet Problem: An Extension to Higher Dimensions of Muskhelishvili Method and Applications ...... 59 A. Cialdea 6.1 Introduction...... 59 6.2 Muskhelishvili’s Method and Its Extension to n ...... 61 6.2.1 Muskhelishvili’s Method ...... 61 6.2.2 Conjugate Differential Forms ...... 61 6.2.3 The Extension to Higher Dimensions of Muskhelishvili Method ...... 63 6.3 The Multiple-Layer Approach ...... 65 References...... 68 7 Bending of Elastic Plates: Generalized Fourier Series Method ...... 71 C. Constanda and D. Doty 7.1 Introduction...... 71 7.2 The Boundary Value Problem ...... 72 7.3 Numerical Example ...... 73 7.4 Graphical Illustrations...... 74 7.5 The Classical Gram–Schmidt Procedure (CGS) ...... 75 7.6 The Modified Gram–Schmidt Procedure (MGS) ...... 77 7.7 The Householder Reflection Procedure (HR) ...... 78 References...... 81 Contents xiii

8 Existence and Uniqueness Results for a Class of Singular Elliptic Problems in Two-Component Domains ...... 83 P. Donato and F. Raimondi 8.1 Introduction...... 83 8.2 Setting of the Problem ...... 84 8.3 A Priori Estimates ...... 87 8.4 Main Results...... 90 References...... 93 9 Fredholmness of Nonlocal Singular Integral Operators with Slowly Oscillating Data ...... 95 G. Fernández-Torres and Yu.I. Karlovich 9.1 Introduction...... 95 9.2 Invertibility Criteria for Wiener Type Functional Operators ...... 98 9.3 Mellin Pseudodifferential Operators ...... 100 9.4 Fredholmness of the Operator N ...... 101 References...... 104 10 Multidimensional Time Fractional Diffusion Equation ...... 107 M. Ferreira and N. Vieira 10.1 Introduction...... 107 10.2 Preliminaries...... 108 10.3 Fundamental Solution of the Multidimensional Time Fractional Diffusion-Wave Equation ...... 110 10.4 Fractional Moments ...... 111 10.5 Graphical Representations of the Fundamental Solution ...... 112 10.5.1 Case n D 1 ...... 112 10.5.2 Case n D 2 ...... 113 10.6 Application to Diffusion in Thermoelasticity ...... 114 References...... 116 11 On Homogenization of Nonlinear Robin Type Boundary Conditions for the n-Laplacian in n-Dimensional Perforated Domains ...... 119 D. Gómez, E. Pérez, A.V. Podol’skii, and T.A. Shaposhnikova 11.1 Introduction...... 119 11.2 Setting of the Problem and Homogenized Problems ...... 122 11.2.1 Table of Homogenized Problems ...... 125 11.3 Preliminary Results ...... 126 11.4 Critical Relation for the Adsorption ...... 128 11.5 Critical Size for Perforations ...... 131 11.6 Extreme Cases ...... 134 References...... 137 12 Interior Transmission Eigenvalues for Anisotropic Media ...... 139 A. Kleefeld and D. Colton 12.1 Introduction...... 139 xiv Contents

12.2 Problem Formulation...... 140 12.3 Boundary Integral Equations ...... 141 12.4 Numerical Results ...... 143 12.5 Summary and Outlook ...... 146 References...... 146 13 Improvement of the Inside-Outside Duality Method ...... 149 A. Kleefeld and E. Reichwein 13.1 Introduction and Motivation ...... 149 13.2 Problem Formulation...... 150 13.3 The Inside-Outside Duality Method ...... 151 13.4 Improvement and Numerical Results...... 152 13.4.1 Lebedev Quadrature ...... 153 13.4.2 Spherical t-Design ...... 154 13.4.3 Numerical Results ...... 155 13.5 Summary and Outlook ...... 157 References...... 158 14 A Note on Optimal Design for Thin Structures in the Orlicz–Sobolev Setting ...... 161 P.A. Kozarzewski and E. Zappale 14.1 Introduction...... 161 14.2 Preliminaries...... 164 14.3 Proof of Theorem 1 ...... 167 References...... 170 15 On the Radiative Conductive Transfer Equation: A Heuristic Convergence Criterion by Stability Analysis ...... 173 C.A. Ladeia, J.C.L. Fernandes, B.E.J. Bodmann, and M.T. Vilhena 15.1 Introduction...... 173 15.2 The Radiative Conductive Transfer Equation in Cylinder Geometry ...... 174 15.3 Solution by the Decomposition Method ...... 175 15.4 A Heuristic Convergence Criterion by Stability Analysis ...... 179 15.5 Numerical Results ...... 180 15.6 Conclusions...... 181 References...... 182 16 An Indirect Boundary Integral Equation Method for Boundary Value Problems in Elastostatics ...... 183 A. Malaspina 16.1 Preliminaries...... 183 16.2 Indirect Method ...... 185 References...... 190 Contents xv

17 An Instability Result for Suspension Bridges ...... 193 C. Marchionna and S. Panizzi 17.1 Introduction...... 193 17.2 The Isoenergetic Poincaré Map and the Asymptotic Behavior of Its Linearization Lk ...... 196 17.3 Some Related Problems and Questions ...... 198 17.4 Numerical Examples ...... 199 References...... 202 18 A New Diffeomorph Conformal Methodology to Solve Flow Problems with Complex Boundaries by an Equivalent Plane Parallel Problem ...... 205 A. Meneghetti, B.E.J. Bodmann, and M.T. Vilhena 18.1 Introduction...... 205 18.2 Coordinate Transformations ...... 205 18.3 Transformation by Invariance ...... 207 18.4 Application to a Navier-Stokes Problem ...... 208 18.5 Application to an Advection-Diffusion Problem ...... 209 18.6 A Numerical Example ...... 209 18.7 Conclusion ...... 213 References...... 214 19 A New Family of Boundary-Domain Integral Equations for the Mixed Exterior Stationary Heat Transfer Problem with Variable Coefﬁcient ...... 215 S.E. Mikhailov and C.F. Portillo 19.1 Introduction...... 215 19.2 Preliminaries...... 216 19.3 Boundary Value Problem ...... 218 19.4 Parametrices and Remainders ...... 218 19.5 Surface and Volume Potentials ...... 219 19.6 Third Green Identities and Integral Relations...... 221 19.7 Boundary-Domain Integral Equation System ...... 222 19.8 Invertibility ...... 224 References...... 225 20 Radiation Conditions and Integral Representations for Clifford Algebra-Valued Null-Solutions of the Iterated Helmholtz Operator ...... 227 D. Mitrea and N. Okamoto 20.1 Introduction...... 227 20.2 Fundamental Solutions ...... 229 20.3 Integral Representations ...... 233 References...... 235 xvi Contents

21 A Wiener-Hopf System of Equations in the Steady-State Propagation of a Rectilinear Crack in an Inﬁnite Elastic Plate ...... 237 A. Nobili, E. Radi, and L. Lanzoni 21.1 Introduction...... 237 21.2 Boundary Conditions...... 240 21.3 Wiener-Hopf Factorization ...... 243 References...... 247 22 Mono-Energetic Neutron Space-Kinetics in Full Cylinder Symmetry: Simulating Power Decrease ...... 249 F.R. Oliveira, B.E.J. Bodmann, M.T. Vilhena, and F. Carvalho da Silva 22.1 Introduction...... 249 22.2 The Model ...... 250 22.3 Variable Separation...... 251 22.4 A Closed Form Solution in Cylinder Geometry ...... 251 22.5 Numerical Results ...... 254 22.6 Conclusion ...... 257 References...... 257 23 Asymptotic Solutions of Maxwell’s Equations in a Layered Periodic Structure ...... 259 M.V. Perel and M.S. Sidorenko 23.1 Introduction...... 259 23.2 Maxwell’s Equations and the Floquet–Bloch Solutions ...... 260 23.3 Two-Scale Solutions of the Maxwell Equations ...... 261 23.4 Boundary Value Problem in a Half-Space ...... 262 23.5 Conclusions...... 264 References...... 264 24 Some Properties of the Fractional Circle Zernike Polynomials ...... 265 M.M. Rodrigues and N. Vieira 24.1 Introduction...... 265 24.2 Preliminaries...... 266 24.3 Fractional Circle Zernike Polynomials ...... 268 24.4 Graphical Representation of Fractional Circle Zernike Polynomials ...... 272 References...... 275 25 Double Laplace Transform and Explicit Fractional Analogue of 2D Laplacian ...... 277 S. Rogosin and M. Dubatovskaya 25.1 Introduction...... 277 25.2 Double Laplace Transforms. Deﬁnition and Main Properties...... 279 Contents xvii

25.3 Application of the Double Laplace Transform to Explicit Fractional Laplacian...... 283 25.3.1 Power Type Fractional Laplacian 4˛;ˇ ...... 283 ˛;ˇIQ 25.3.2 Skewed Fractional Laplacian 4C in the Quarter-Plane ...... 284 25.4 Discussion and Outlook ...... 291 References...... 292 26 Stability of the Laplace Single Layer Boundary Integral Operator in Sobolev Spaces ...... 293 O. Steinbach 26.1 Introduction...... 293 26.2 Strong Domain Variational Formulation ...... 295 26.3 Ultra-Weak Domain Variational Formulation...... 298 26.4 Single Layer Potential...... 299 References...... 303 27 Spectral Lanczos’ Tau Method for Systems of Nonlinear Integro-Differential Equations ...... 305 P.B. Vasconcelos, J. Matos, and M.S. Trindade 27.1 Introduction...... 305 27.2 Preliminaries...... 306 27.3 The Tau Method for Integro-Differential Problems ...... 308 27.4 Nonlinear Approach for Integro-Differential Problems ...... 309 27.5 Contributions to Stability ...... 310 27.6 Numerical Results ...... 311 27.7 Conclusions...... 314 References...... 314 28 Discreteness, Periodicity, Holomorphy, and Factorization ...... 315 V.B. Vasilyev 28.1 Introduction...... 315 28.2 Discreteness ...... 316 28.3 Periodicity ...... 317 28.4 Holomorphy ...... 318 28.5 Factorization ...... 318 28.5.1 Conical Case ...... 320 References...... 324 29 Modes Coupling Seismic Waves and Vibrating Buildings: Existence ...... 325 D. Volkov and S. Zheltukhin 29.1 Introduction...... 325 xviii Contents

29.2 Low Frequency Analysis...... 327 29.3 High Frequency Asymptotic Analysis ...... 330 References...... 333

Index ...... 335 List of Contributors

Nadia M. Abusag Nottingham Trent University, Nottingham, UK Mario Ahues Blanchait University of Lyon, Saint–Étienne, France Éder L. de Albuquerque University of Brasilia, Brasilia, Brazil Filomena D. d’Almeida University of Porto, Porto, Portugal Francesco Altomare University of Bari, Bari, Italy Antônio C.A. Alvim Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil Tsegaye G. Ayele Addis Ababa University, Addis Ababa, Ethiopia Luiz F.F.C. Barcellos Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Ricardo C. Barros State University of Rio de Janeiro, Nova Friburgo, RJ, Brazil Miriam Belmaker The University of Tulsa, Tulsa, OK, USA Elvise Berchio Polytechnic University of Torino, Torino, Italy Bardo E.J. Bodmann Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Davide Buoso Polytechnic University of Torino, Torino, Italy Lucas Campos University of Brasilia, Brasilia, Brazil Haroldo F. Campos Velho National Institute for Space Research, São José dos Campos, SP, Brazil Luis P. Castro University of Aveiro, Aveiro, Portugal David J. Chappell Nottingham Trent University, Nottingham, UK Nadia Chuzhanova Nottingham Trent University, Nottingham, UK

xix xx List of Contributors

Alberto Cialdea University of Basilicata, Potenza, Italy David L. Colton University of Delaware, Newark, DE, USA Rafael Company Polytechnic University of Valencia, Valencia, Spain Christian Constanda The University of Tulsa, Tulsa, OK, USA Jonathan J. Crofts Nottingham Trent University, Nottingham, UK Jesús Pérez Curbelo State University of Rio de Janeiro, Nova Friburgo, RJ, Brazil Fateme Daburi Farimani Ferdowsi University of Mashhad, Mashhad, Iran Panagiotis Dimitrakopoulos The University of Maryland, College Park, MD, USA Patrizia Donato University of Rouen Normandie, Saint-Étienne-du-Rouvray, France Dale Doty The University of Tulsa, Tulsa, OK, USA Maryna V. Dubatovskaya The Belarusian State University, Minsk, Belarus Tamirat T. Dufera Adama Science and Technology University, Adama, Ethiopia Mahdi A. Esfahani Ferdowsi University of Mashhad, Mashhad, Iran Carlos E. Espinosa Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Mohamed Fakharany Tanta University, Tanta, Egypt Julio C.L. Fernandes Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Rosário Fernandes University of Minho, Braga, Portugal Gustavo Fernández-Torres National Autonomous University of México, Ciudad de México, Mexico Milton Ferreira Polytechnic Institute of Leiria, Leiria, Portugal Fahin Forouzanfar The University of Tulsa, Tulsa, OK, USA Saulo R. Freitas NASA Goddard Space Flight Center, Greenbelt, MD, USA Igor C. Furtado Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Carlos R. García Institute of Technology and Applied Sciences, La Habana, Cuba Filippo Gazzola Polytechnic University of Milano, Milano, Italy Delﬁna Gómez University of Cantabria, Santander, Spain Rita C. Guerra University of Aveiro, Aveiro, Portugal Paul J. Harris University of Brighton, Brighton, UK List of Contributors xxi

Gabriel Hattori Durham University, Durham, UK Randy D. Hazlett The University of Tulsa, Tulsa, OK, USA Adino Heimlich Nuclear Engineering Institute, Rio de Janeiro, RJ, Brazil Lucas Jódar Polytechnic University of Valencia, Valencia, Spain Hanane Kaboul University of Lyon, Saint–Étienne, France Saleh Kargarrazi The Royal Institute of Technology, Kista, Sweden Abolfazl Karimpour Iran University of Science and Technology, Tehran, Iran Ali Karimpour Ferdowsi University of Mashhad, Mashhad, Iran Mostafa Karimpour Ferdowsi University of Mashhad, Mashhad, Iran Yuri I. Karlovich Autonomous State University of Morelos, Cuernavaca, Morelos, México Sam H. Kettle Durham University, Durham, UK Andreas Kleefeld Forschungszentrum Jülich GmbH, Jülich, Germany Kianoush Kompany Virginia Tech, Blacksburg, VA, USA Gene Kouba Chevron Energy Technology Company (Retired), Houston, TX, USA Piotr Kozarzewski University of Warsaw, Warsaw, Poland Cibele A. Ladeia Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Luca Lanzoni University of San Marino, San Marino, San Marino Sergio Q. Bogado Leite National Nuclear Energy Commission, Rio de Janeiro, RJ, Brazil Vita Leonessa University of Basilicata, Potenza, Italy Angelica Malaspina University of Basilicata, Potenza, Italy Clelia Marchionna Polytechnic University of Milano, Milano, Italy Tamires Marotto North Fluminense State University, Macaé, RJ, Brazil Rebecca Martin Nottingham Trent University, Nottingham, UK Aquilino S. Martinez Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil José M.A. Matos Politechnic School of Engineering, Porto, Portugal André Meneghetti Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Abdelaziz Mennouni University of Batna 2, Batna, Algeria xxii List of Contributors

Sergey E. Mikhailov Brunel University West London, Uxbridge, UK Dorina Mitrea University of Missouri, Columbia, MO, USA Ram Mohan The University of Tulsa, Tulsa, OK, USA Mirella Cappelletti Montano University of Bari, Bari, Italy Hunghu Nguyen The University of Tulsa, Tulsa, OK, USA Andrea Nobili University of Modena and Reggio Emilia, Modena, Italy Nicholas H. Okamoto University of Missouri, Columbia, MO, USA Fernando R. Oliveira Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Stefano Panizzi University of Parma, Parma, Italy Maria V. Perel St. Petersburg State University, St. Petersburg, Russia Alvaro M.M. Peres North Fluminense State University, Macaé, RJ, Brazil Maria Eugenia Pérez University of Cantabria, Santander, Spain Gary Phillips University of Brighton, Brighton, UK Adolfo P. Pires North Fluminense State University, Macaé, RJ, Brazil Alexander V. Podol’skii Moscow State University, Moscow, Russia Carlos F. Portillo Oxford Brookes University, Wheatley, UK Hamid-Reza Pourreza Ferdowsi University of Mashhad, Mashhad, Iran Maryam Pourreza Sharif University of Technology, Tehran, Iran Enrico Radi University of Modena and Reggio Emilia, Reggio Emilia, Italy Federica Raimondi University of Salerno, Fisciano, SA, Italy Habib Rajabi Mashhadi Ferdowsi University of Mashhad, Mashhad, Iran Ioan Ra¸sa Technical University of Cluj-Napoca, Cluj-Napoca, Romania Elisabeth Reichwein Heinrich-Heine-Universität, Düsseldorf, Germany Manuela Rodrigues University of Aveiro, Aveiro, Portugal Sergei V. Rogosin The Belarusian State University, Minsk, Belarus Heloisa M. Ruivo National Institute for Space Research, São José dos Campos, SP, Brazil Renata S.R. Ruiz National Institute for Space Research, São José dos Campos, SP, Brazil Javad Sadeh Ferdowsi University of Mashhad, Mashhad, Iran List of Contributors xxiii

Aleksey V. Setukha Lomonosov Moscow State University, Moscow, Russia Tatiana A. Shaposhnikova Moscow State University, Moscow, Russia Ovadia Shoham The University of Tulsa, Tulsa, OK, USA Mikhail S. Sidorenko Ioffe Institute, St Petersburg, Russia Fernando C. Silva Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil OdairP.daSilva State University of Rio de Janeiro, Nova Friburgo, Brazil Adel Soheili Ferdowsi University of Mashhad, Mashhad, Iran Stanislav L. Stavtsev Institute of Numerical Mathematics of the Russian Academy of Sciences, Moscow, Russia Olaf Steinbach Graz University of Technology, Graz, Austria P. Stpiczynski´ Maria Curie-Skłodowska University, Lublin, Poland Jon Trevelyan Durham University, Durham, UK Marcelo S. Trindade University of Porto, Porto, Portugal Nguyen M. Tuan National University of Viet Nam, Hanoi, Vietnam Paulo B. Vasconcelos University of Porto, Porto, Portugal Vladimir B. Vasilyev National Belgorod Research State University, Belgorod, Russia Jenny Venton University of Brighton, Brighton, UK Bruno J. Vicente North Fluminense State University, Macaé, RJ, Brazil Nelson Vieira University of Aveiro, Aveiro, Portugal Marco T.B.M. Vilhena Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Darko Volkov Worcester Polytechnic Institute, Worcester, MA, USA Maria E.S. Welter National Institute for Space Research, São José dos Campos, SP, Brazil Elvira Zappale University of Salerno, Fisciano (SA), Italy Sergey Zheltukhin Riﬁniti, Inc., Boston, MA, USA Barbara Zubik-Kowal Boise State University, Boise, ID, USA Chapter 1 An L1-Product-Integration Method in Astrophysics

M. Ahues Blanchait and H. Kaboul

1.1 Introduction

We consider a Banach space X.LetT be the integral operator deﬁned by Z b 8x 2 X; 8s 2 Œa; b; Tx.s/ WD L.s; t/H.s; t/x.t/ dt; a where .s; t/ 7! H.s; t/ is not smooth. For z not in the spectrum of T, and any y in X we tackle the problem

Find ' 2 X s.t. .T zI/' D y; where I denotes the identity operator on X. The solution ' will be approximated through the exact solution 'n of a ﬁnite rank equation

.Tn zI/'n D y:

To do so, we propose a product-integration scheme in X WD L1.Œa; b; ¼/.

M.A. Blanchait () • H. Kaboul Camille Jordan Institute, University of Lyon, Cedex 2, France e-mail: [email protected]; [email protected]

1.2 A Product-Integration Method in L1.Œa; b; ¼/ Z b 1 In the sequel, kxk1 WD jx.s/j ds is the norm in L .Œa; b; ¼/. The subordinated a 1 operator norm is also denoted by k:k1. The oscillation of a function x in L .Œa; b; ¼/, relative to a h 2 is Z b w1.x; h/ WD sup jx.v C u/ x.v/j dv; juj2Œ0;jhj a where x.t/ WD 0 for t … Œa; b. The modulus of continuity of a continuous function on Œa; b is

w.x; h/ WD sup jx.u/ x.v/j: u;v2Œa;b;juvjÄjhj

The modulus of continuity of f on Œa; b Œa; b is

w2.f ; h/ WD sup jf .u/ f .v/j: u;v2Œa;b2;kuvkÄjhj

For x 2 L1.Œa; b; ¼/

lim w1.x; h/ D 0: h!0

1 T is a compact bounded linear operator from L .Œa; b; ¼/ into itself and .Tn/n2 is a collectively compact approximation of T. The compactness in L1.Œa; b; ¼/ relies on the Kolmogorov-Riesz-Fréchet theorem. We assume that (P1) L 2 C0.Œa; b Œa; b; ¼/. Let

cL WD sup jL.s; t/j: .s;t/2Œa;b2

(P2) H veriﬁes: Z b (P2.1) cH WD sup jH.s; t/j ds is ﬁnite. t2Œa;b a (P2.2) lim wH.h/ D 0; h!0 where Z b wH.h/ WD sup jHQ .s C h; t/ HQ .s; t/j ds; t2Œa;b a 1 Product Integration Method in Astrophysics 3

and H.s; t/ for s 2 Œa; b; HQ .s; t/ WD 0 for s … Œa; b:

Lemma 1

lim .H; h/ D 0; h!0C where Z b .H; h/ WD sup jH.s; t/j ds: t2Œa;b bh

Theorem 1 Under the assumptions (P1) and (P2), the operator T is linear from L1.Œa; b; ¼/ into itself and compact in L1.Œa; b; ¼/. 1 Let n be a uniform grid with mesh size hn.Forx 2 L .Œa; b; ¼/, Z 1 1 tn;i t 7! Qn.x; s; t/ WD .tn;i t/L.s; tn;i1/ C .t tn;i1/L.s; tn;i/ x.u/ du hn hn tn;i1

1 for i D 1;:::;n, and t 2 Œtn;i1; tn;i, and 8x 2 L .Œa; b; ¼/, 8s 2 Œa; b, Z b Tnx.s/ WD Qn.x; s; t/H.s; t/ dt a Xn D cn;iwn;i.s/; iD1 where, for i D 1;:::;n; Z 1 tn;i cn;i WD x.u/ du; hn tn;i1 and Z tn;i wn;i.s/ WD Qn.1; s; t/H.s; t/ dt: tn;i1

Lemma 2 For i D 1;:::;n, Z b jwn;i.s/j ds Ä hncLcH: a 4 M.A. Blanchait and H. Kaboul

For h 2 C; Z b jwn;i.s/j ds Ä hncL.H; h/; bh Z bˇh ˇ ˇ ˇ wn;i.s C h/ wn;i.s/ ds Ä hncHw2.L; h/ C hncLwH.h/: a

Lemma 3 For x 2 L1.Œa; b; ¼/, Z Xn tn;i jx.u/ cn;ij du Ä 2w1.x; hn/: iD1 tn;i1

For t 2 Œa; b; ˇ ˇ ˇ ˇ Qn.1; s; t/ L.s; t/ Ä w2.L; hn/:

1 Theorem 2 Tn is a compact linear operator from L .Œa; b; ¼/ into itself and .Tn/n2 is a collectively compact approximation to T.

Lemma 4 Let z 2 re.T/. For n large enough, Tn zI is invertible and it exists Mz >0such that

1 k.Tn zI/ k1 Ä cz:

Proof It is a consequence of the collectively compact convergence (see [An71]). Theorem 3 For z not in the spectrum of T, and under hypotheses (P1) and (P2), for n large enough, 'n is uniquely deﬁned and

k' 'nk1 Ä czcH .k'k1w2.L; hn/ C 2cLw1.'; hn// :

In the transfer equation, the kernel H is of convolution type: a D 0, b D 1 and H.s; t/ D g.js tj/, where • lim g.s/ DC1, s!0C • g 2 C0.0; 1; / \ L1.Œ0; 1; /, • g 0 and g is a decreasing function in 0; 1.

1.3 Iterative Reﬁnement

The approximate solution can be written as 'n D Gn.z/y, where Gn.z/ is an approximate inverse of T zI, and its accuracy may be improved using an iterative reﬁnement scheme: 1 Product Integration Method in Astrophysics 5

.0/ . / ; xn WD Gn z y .kC1/ .0/ . . /. // .k/; xn WD x C I Gn z T zI xn such as 1 Scheme A (Atkinson): Gn.z/ WD Rn.z/ WD .Tn zI/ , 1 Scheme B (Brakhage): G .z/ WD .R .z/T I/, n z n . / 1 . . / / Scheme C (Titaud): Gn z WD z TRn z I . Theorem 4 If h ! 0C, then

. / 3 2 . ; / 2 2 . / 2 2. ; / 0; e h WD CHCLw L h C CHCLwH h C CHCL H h ! and there exist az >0,bz >0and cz >0independent of n such that: For Scheme A:

.2`1/ .2`/ kxn 'k kxn 'k Ä .b e.h //`; Ä a .b e.h //`: k'k z n k'k z z n

For Scheme B:

.k/ kxn 'k c kC1 Ä z e.h / : k'k z n

For Scheme C:

.k/ kxn 'k c c k Ä z z e.h / : k'k z z n

1.4 Numerical Evidence

The approximate equation is

n Z X 1 tn;j 8s 2 Œa; b; w ; .s/ ' .u/ du z' .s/ D y.s/: n j h n n iD1 n tn;j1

If we compute the local mean over Œtn;i1; tn;i; i D 1;:::;n, we get a linear system .A zI/x D d, where Z 1 tn;i A.i; j/ WD wn;j.s/ ds; i; j D 1;:::;n; hn tn;i1 6 M.A. Blanchait and H. Kaboul Z 1 tn;i d.i/ WD y.s/ ds; i D 1;:::;n; hn tn;i1 Z 1 tn;i x.i/ WD 'n.s/ ds; i D 1;:::;n; hn tn;i1 and

1 Xn Á ' .s/ D w ; .s/x.i/ y.s/ : n z n j iD1

The quality of 'n is estimated through the relative residual:

k.T zI/'n yk1 r.'n/ WD : kyk1

The radiative transfer problem describes the energy conserved by a beam radiation traveling. Let be the optical width of the medium (see [ChEtAl07]). The equation is Z $.s/ E1.js tj/'.t/ dt '.s/ D y.s/; 2 0 where E1 is the ﬁrst integral exponential function, the albedo is $.s/ D 0:7 exp.s/, and 0:3 for s 2 Œ0; 50Œ; y.s/ D 0 for s 2 Œ50; 100:

The relative residual associated with 'n has been computed both by the Kantorovich projection method and by the product-integration method. Results are shown in Table 1.1. Since for n 100, the computation of 'n is too costly, we use a reﬁnement scheme to compute a better approximate solution. The number of iterations needed to get kRelative residualkÄ1012 is shown in Table 1.2.

Table 1.1 Relative residuals n Projection method Product-integration method 10 0.0267 0.0172 20 0.0252 0.0145 50 0.0151 0.0075 1 Product Integration Method in Astrophysics 7

Table 1.2 Number of Iterations to get kRelative residualkÄ1012 with n D 100 Scheme A Scheme B Scheme C 124 9 9

Acknowledgements This research has been partially supported by the Indo French Center for Applied Mathematics (IFCAM).

References

[An71] Anselone, P.M.: Collectively Compact Operator Approximation Theory and Appli- cation to Integral Equations. Prentice-Hall, Englewoods Cliffs, NJ (1971) [ChEtAl07] Chevallier, L., Pelkowski, J., Rutily, B.: Exact results in modeling planetary atmospheres–I. Gray atmospheres. J. Quant. Spectrosc. Radiat. Transf. 104, 357 (2007) Chapter 2 Differential Operators and Approximation Processes Generated by Markov Operators

F. Altomare, M. Cappelletti Montano, V. Leonessa, and I. Ra¸sa

2.1 Introduction

In recent years several investigations have been devoted to the study of large classes of (mainly degenerate) initial-boundary value evolution problems in connection with the possibility to obtain a constructive approximation of the associated positive C0-semigroups by means of iterates of suitable positive linear operators which also constitute approximation processes in the underlying Banach function space. Usually, as a consequence of a careful analysis of the preservation properties of the approximating operators, such as monotonicity, convexity, Hölder continuity, and so on, it is possible to infer similar preservation properties for the relevant semigroups and, in turn, some spatial regularity properties of the solutions to the evolution problems (see, e.g., [CaEtAl99, AtCa00, Ma02, AlEtAl07, AlLe09], [AlCa94, Chapter 6] and the references therein). More recently, by continuing along these directions, we started a research project in order to investigate the possibility of associating to a given Markov operator on the Banach space C.K/ of all real functions deﬁned on a convex compact subset K of d (d 1) some classes of differential operators as well as some suitable positive approximation processes. The main aim is to investigate whether these differential operators are generators of positive semigroups and whether the semigroups can

F. Altomare • M. Cappelletti Montano University of Bari, Bari, Italy e-mail: [email protected]; [email protected] V. Leonessa () University of Basilicata, Potenza, Italy e-mail: [email protected] I. Ra¸sa Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania e-mail: [email protected]

© Springer International Publishing AG 2017 9 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_2 10 F. Altomare et al. be approximated by iterates of the approximation processes themselves. By means of a qualitative study of the approximation processes, the approximation formulas could guarantee a similar qualitative analysis of the positive semigroups and, consequently, of the solutions to the evolution equations governed by them. In this survey paper we report some of the main ideas and results we have developed in this respect and which are documented in [AlEtAl14, AlEtAl14a, AlEtAl14b, AlEtAl16a, AlEtAl16b]. The differential operators considered within the framework of the theory fall into classes of operators of wide interest in the theory of evolution equations and in models of population dynamics and mathematical ﬁnance. The generation problems for these differential operators have been also studied with other methods which, however, do not allow to get spatial regularity properties of the solutions as well as information about their asymptotic behavior whereas these aspects are successfully obtained with the methods of approximation by positive operators. Furthermore, the involved approximation processes are inspired by some classical ones and, among other things, they generalize the Bernstein operators and the Kantorovich operators in all one-dimensional and multi-dimensional convex domains on which the latter have been considered. These approximation processes seem to have an interest in their own also for the approximation of continuous functions and, in some cases, of p-th power integrable functions. For these reasons their study has been also deepened from several points of view of the approximation theory. The paper contains some noteworthy examples which offer a short outline of the possible application of the theory and show that diverse differential problems scattered in the literature can be encompassed in the present unifying approach (see, e.g., [CeCl01, MuRh11, Ta14]).

2.2 Canonical Elliptic Second-Order Differential Operators and Bernstein-Schnabl Operators

From now on we shall ﬁx a convex compact subset K of the real Euclidean space d (d 1) with non-empty interior int.K/, and a Markov operator T W C.K/ ! C.K/ on the space C.K/ of all real continuous functions on K, i.e. T is a positive linear operator on C.K/ such that T.1/ D 1, 1 being the constant function of value 1 on K. By Riesz representation theorem it is known that, for every x 2 K, there exists a T . / (unique) probability Borel measure Q x on K such that, for every f 2 C K , Z . /. / T : T f x D fdQ x K

Then, for every n 1, we deﬁne the n-th Bernstein-Schnabl operator Bn associated with T by setting, for every f 2 C.K/ and x 2 K, Z Z Â Ã x1 ::: x . /. / C C n T . / T . /: Bn f x WD f d Q x x1 d Q x xn K K n 2 Approximation Processes Generated by Markov Operators 11

For every n 1, Bn is a positive linear operator from C.K/ into C.K/, Bn.1/ D 1 and hence, by positivity, kBnkD1. Moreover, B1 D T. If, in addition, the Markov operator T satisﬁes the assumption

T.h/ D h for every h 2fpr1;:::;prdg (2.1)

th where, for each i D 1;:::;d, pri stands for the i coordinate function on K, i.e. pri.x/ D xi for every x D .x1;:::;xd/ 2 K, then the sequence .Bn/n1 is an approximation process on C.K/, i.e. for every f 2 C.K/, limn!1 Bn.f / D f uniformly on K (see [AlEtAl14a, Theorem 3.1]; see also [AlEtAl14, Theorem 3.2.1]). The class of Bernstein-Schnabl operators associated with T contains several examples of well-known sequences of operators as particular cases. In fact they generalize the classical Bernstein operators on the unit interval, on multi-dimensional simplices and hypercubes, and they share with them several preservation properties which are investigated in [AlEtAl14, Chapter 3] and [AlCa94, Chapter 6] (see also [AlEtAl14a] and the references therein). Among the properties fulﬁlled by the Bn’s, we recall here that, for every m 1, all the Bernstein-Schnabl operators leave invariant the space Pm.K/ of (restriction to K of all) polynomials of degree at most m, under the additional hypothesis

T.Pm.K// Pm.K/ (2.2) for every m 1 (see [AlEtAl14a, Theorem 3.2]; see also [AlEtAl14, Lemma 4.1.1]). Moreover, for every u 2 C2.K/, the following asymptotic formula holds:

lim n.Bn.u/ u/ D WT .u/ uniformly on K; (2.3) n!1 where WT is the elliptic second-order differential operator deﬁned as

1 Xd @2u W .u/ WD ˛ (2.4) T 2 ij @x @x i;jD1 i j and the coefficients ˛ij, for each i; j D 1;:::;d and x 2 K,aregivenby˛ij.x/ WD T.priprj/.x/ .priprj/.x/ (see [AlEtAl14a, Theorem 4.2]; see also [AlEtAl14, Theorem 4.1.5]). WT is referred to as the canonical elliptic second-order differential operator associated with T. Operators (2.4) are of concern in the study of several diffusion problems arising in biology, financial mathematics, and other fields. As a matter of fact the study of such kind of differential operator presents some difficulties if tackled with the methods of the theory of PDEs. First of all the boundary @K of K is generally non- smooth due to the presence of possible sides and corners. Moreover, WT degenerates on the set @T K of all interpolation points for T given by @T K WD fx 2 K j T.f /.x/ D f .x/ for every f 2 C.K/g; which contains the extreme points of K by virtue of (2.1). 12 F. Altomare et al.

Now consider the following initial-boundary value problem associated with the 2 2 couple .WT ; C .K// and the initial datum u0 belonging to a suitable subset of C .K/: 8 <ˆ @u 1 Pd @2u .x; t/ D ˛ij.x/ .x; t/ x 2 K; t 0I @t 2 ; 1 @xi@xj (2.5) :ˆ i jD u.x;0/D u0.x/ x 2 K:

The possibility to approximate (or to reconstruct) the solutions to (2.5) lies in the fact that, under hypotheses (2.1) and (2.2) on the Markov operator T, by applying a Trotter-Schnabl-type result (see [AlEtAl14, Corollary 2.2.3 and Remark 2.2.4]; see also [AlEtAl09, Theorem 2.1]), we get that 2 Theorem 1 The operator .WT ; C .K// is closable and its closure .AT ; D.AT // generates a Markov semigroup .T.t//t0 on C.K/ such that, if t 0 and .k.n//n1 is a sequence of positive integers satisfying lim k.n/=n D t, then n!1

T.t/.f / D lim Bk.n/.f / uniformly on K (2.6) n!1 n

. / k.n/ . / for every f 2 C K (hereS Bn denotes the iterate of Bn of order k n ). Moreover, the . / . / . / 2. / subalgebra P1 K WD m1 Pm K of C K , and hence C K as well, is a core for .AT ; D.AT //. The approximation formula (2.6) will be brieﬂy referred by saying that the sequence .Bn/n1 is a strongly admissible sequence for the semigroup .T.t//t0. From the above theorem it follows that we can solve the abstract Cauchy problem associated with .AT ; D.AT // and the initial datum u0 2 D.AT / 8 < du .x; t/ D AT .u.; t//.x/ x 2 K; t 0I : dt (2.7) u.x;0/D u0.x/ x 2 K;

2 which turns into the particular problem (2.5) when u0 is in C .K/, since AT D WT on C2.K/, obtaining for it an approximation formula for the (unique) solution, i.e.

k.n/ u.x; t/ WD T.t/.u0/.x/ D lim B .u0/ uniformly on K: (2.8) n!1 n

For more details on semigroup theory, we refer the reader to [EnNa00]orto [AlEtAl14, Chapter 2]. From (2.8) we can derive both numerical and qualitative information about the solutions of the Cauchy problems of kind (2.5) from the study of operators Bn. Below we list some spatial regularity properties of the solutions to (2.7) which may be inferred by some preservation properties of the Bn’s by means of (2.8). 2 Approximation Processes Generated by Markov Operators 13

Theorem 2 Under the same assumptions of Theorem 1, the following statements hold true: (i) Given M 0 and 0<˛Ä 1, let Lip.M;˛/ be the space of all Hölder continuous functions on K with exponent ˛ and Hölder constant M. If, in addition, T.Lip.1; 1// Lip.1; 1/ and u0 2 Lip.M;˛/, then u.; t/ 2 Lip.M;˛/ for every t 0,M 0 and 0<˛Ä 1. (ii) Suppose that T maps continuous convex functions into (continuous) convex functions and that the quantity ZZ Â Ã s C t T T .f x; y/ B2.f /.x/ B2.f /.y/ 2 f d .s/d .t/ I WD C 2 Q x Q x K2

is positive for every f 2 C.K/ and x; y 2 K. If u0 2 D.AT / is convex, then u.; t/ is convex for every t 0. We conclude this section by presenting several examples of Markov operators to which the previous theorems apply. Example 1 Assume that d 2 and that @K is an ellipsoid, i.e. 8 9 < Xd = d . / . /. / 1 ; K D :x 2 j Q x x WD rij xi xi xj xj Ä ; i;jD1

d where .rij/i;jD1;:::;d is a real symmetric and positive-deﬁnite matrix and x 2 .Let L be a strictly elliptic differential operator of the form

Xd @2u.x/ L.u/.x/ D c .u 2 C2.int.K//; x 2 int.K// ij @x @x i;jD1 i j associated with a real symmetric and positive matrix .cij/1Äi;jÄd and denote by TL W C.K/ ! C.K/ the Poisson operator associated with L. Thus, for every f 2 C.K/, TL.f / denotes the unique solution to the Dirichlet problem ( Lu D 0 on int.K/; u 2 C.K/ \ C2.int.K//I u D f on @K:

Note that TL is a Markov operator (in particular a Markov projection) satisfying (2.1). Moreover, also (2.2) is veriﬁed.

The differential operator WTL associated with TL, brieﬂy denoted by WL,is given by 8 ˆ 1 Q.x x/ ˆ L.u/.x/ if x 2 int.K/I < Pd WL.u/.x/ WD 2 rijcij ˆ ; 1 :ˆ i jD 0 if x 2 @K

(u 2 C2.K/, x 2 K). 14 F. Altomare et al.

In particular, if K is the closed ball (with respect to kk2) with center the origin of d and radius 1 and if L is the laplacian , then

8 2 Z < 1 kxk2 f .z/ d.z/ if kxk2 <1I . /. / d T f x WD : d @K kz xk2 f .x/ if kxk2 D 1

d (f 2 C.K/, x 2 K), where d and denote the surface area of the unit sphere in and the surface measure on @K, resp., and, for every u 2 C2.K/ and x 2 K, 8 2 < 1 kxk2 .u/.x/ x 2 <1 . /. / 2 if k k I W u x WD : d 0 if kxk2 D 1:

An explicit expression for the Bernstein-Schnabl operators associated with TL can be found in [AlEtAl14, Subsection 3.1.4]. Example 2 Consider the d-dimensional simplex ( ) Xd d Kd WD .x1;:::;xd/ 2 j xi 0 for every i D 1;:::;d and xi Ä 1 iD1 and the canonical projection Td on Kd deﬁned by ! Xd Xd Td.f /.x/ WD 1 xi f .v0/ C xif .vi/ iD1 iD1

.f 2 C.Kd/; x D .x1;:::;xd/ 2 Kd/, where v0 WD .0;:::;0/, v1 WD .1;0;:::;0/, :::, vd WD .0;:::;0;1/are the vertices of the simplex. Then Td is a Markov operator which satisﬁes (2.1) and (2.2) and the relevant differential operator WTd associated with Td is given by

1 Xd @2u X @2u WT .u/.x/ D xi.1 xi/ .x/ xixj .x/ d 2 @x2 @x @x iD1 i 1Äi

. 2. /; . ;:::; / / u 2 C Kd x D x1 xd 2 Kd . Note that the coefﬁcients of WTd vanish on the vertices of the simplex.

The operator WTd falls into the class of the so-called Fleming-Viot operators which occur in the description of a stochastic process associated with a diffusion approximation of a gene frequency model in population genetics; more recently, they have been object of investigation by several authors. Also in this case we have an explicit expression for the relevant operators Bn (see [AlEtAl14, formula (3.1.18)]). 2 Approximation Processes Generated by Markov Operators 15

Example 3 Let S W C.Kd/ ! C.Kd/ be the Markov operator deﬁned by 8 0 1 ˆ ˆ @ x1 A x1 ˆ 1 .0; 2;:::; / ˆ Pd f x xd C Pd ˆ 1 1 <ˆ xi xi Â iD2 Ã iD2 . /. / Pd Pd S f x WD ˆ 1 ; ;:::; 1 ˆ f xi x2 xd if xi ¤ I ˆ iD2 iD2 ˆ Pd :ˆ f .0; x2;:::;xd/ if xi D 1 iD2

.f 2 C.Kd/; x D .x1;:::;xd/ 2 Kd/. S veriﬁes (2.1) and (2.2) and in this case the differential operator associated with S is deﬁned as ! 1 Xd @2u WS.u/.x/ D x1 1 xi .x/ 2 @x2 iD1 1

. 2. /; . ;:::; / / u 2 C Kd x D x1 xd 2 Kd . Note that WS degenerates on the faces Pd fx D .x1;:::;xd/ 2 Kd j x1 D 0g and x D .x1;:::;xd/ 2 Kd j xi D 1 . iD1 Example 4 Consider the convex combination of the above operators, that is the Markov operator V WD .Td C S/=2. Then V satisﬁes (2.1) and (2.2) and the differential operator associated with it becomes ! 1 Xd @2u WV .u/.x/ D 2x1.1 x1/ x1 xi .x/ 4 @x2 iD2 1

1 Xd @2u 1 X @2u C xi.1 xi/ .x/ xixj .x/ 4 @x2 4 @x @x iD2 i 1Äi

2 .u 2 C .Kd/; x D .x1;:::;xd/ 2 Kd/. Observe that WV degenerates on the vertices of Kd as well. In this case Bn’s are deﬁned as in the Subsection 3.1.6 of [AlEtAl14]. d Example 5 Let Qd WD Œ0; 1 , d 1, and for every i D 1;:::;d consider a Nd Markov operator Ui on C.Œ0; 1/ satisfying (2.1) and (2.2). If U WD Ui is the iD1 tensor product of the family .Ui/1ÄiÄd, then U is a Markov operator on C.Qd/ satisfying (2.1) and (2.2). The differential operator in this case has the following form:

1 Xd @2u WU.u/.x/ D ˛i.x/ .x/; 2 @x2 iD1 i 16 F. Altomare et al.

. 2. /; . ;:::; / / ˛ . / . /. / 2 .1 /; u 2 C Qd x D x1 xd 2 Qd , where i x WD Ui e2 xi xi Ä i Ä d 2 e2 being the function e2.t/ D t for each t 2 Œ0; 1. Bernstein-Schnabl operators on hypercubes are given by formula (3.1.30) of [AlEtAl14].

2.3 Other Classes of Differential Operators and Approximation Processes

In this section we discuss other classes of differential operators which are pre- generators of Markov semigroups along with the relevant strongly admissible sequences of positive operators. The differential operators are multiplicative and additive perturbations of the operator WT and the relevant approximation processes can be obtained by means of suitable modiﬁcation of Bernstein-Schnabl operators. Multiplicative Perturbations and Lototsky-Schnabl Operators Given a Markov operator T on C.K/ satisfying assumption (2.1), a new Markov operator U can be constructed by setting U WD T C .1 /I; where 2 C.K/, 0 Ä Ä 1, I being the identity operator of C.K/.

The Bernstein-Schnabl operators Bn;U associated with U are referred to as the Lototsky-Schnabl operators associated with T and and they are simply denoted by Bn;, n 1. Actually, if f 2 C.K/, x 2 K and n 1, 8 ! ˆ Xn < n k nk .x/ .1 .x// Bk.fx;k=n/.x/ if x … @T K; Bn;.f /.x/ WD k :ˆ kD0 f .x/ if x 2 @T K; . / k 1 k where fx;k=n t D f n t C n x (t 2 K). It is possible to show that, on one hand, the multiplicative perturbation WT of the differential operator WT is linked to the operators Bn; by an asymptotic formula like (2.3), and on the other 2 hand that the couple .WT ; C .K// pre-generates a Markov semigroup for which a formula similar to (2.6) involving iterates of Lototsky-Schnabl operators holds under the same hypotheses (2.1) and (2.2)onT. For all details we refer the reader to [AlEtAl14, Section 5.1]. Special additive Perturbation and Generalized Kantorovich Operators In the recent paper [AlEtAl16a] we introduce and study a new sequence of positive linear operators acting on C.K/ (and, in some particular cases, also in other function spaces). Namely, if T is a Markov operator on C.K/ satisfying (2.1), if a is a real positive number, and if .n/n1 is a weakly convergent sequence of probability Borel measures on K, we deﬁne the positive linear operator Cn by setting, for every x 2 K and f 2 C.K/, Z Z Â Ã x1 ::: x ax 1 . /. / C C n C nC T . / T . / . /: Cn f x D f d Q x x1 d Q x xn d n xnC1 K K n C a 2 Approximation Processes Generated by Markov Operators 17

Through an asymptotic formula like (2.3) such operators are connected with the following class of differential operators which turn into particular additive 2 perturbations of WT which are deﬁned by setting, for every u 2 C .K/ and x 2 K,

Xd @u V .u/.x/ WD W .u/.x/ C a.b pr / ; T T i i @x iD1 i where b D .b1;:::;bd/ 2 K is the barycenter of the weak limit of .n/n1 (see 2 [AlEtAl16b]). Then .VT ; C .K// is a pre-generator of a Markov semigroup for which the sequence .Cn/n1 is a strongly admissible sequence. In some particular cases, the same result holds true also in Lp-spaces. Additive Perturbation and Modiﬁed Bernstein-Schnabl Operators Consider the complete differential operator

Xd @u Z .u/.x/ D W .u/.x/ C ˇ .x/ C .x/u.x/.u 2 C2.K/; x 2 K/; T T i @x iD1 i

ˇ.x/ with ˇ ; 2 C.K/ for which there exists n0 1 such that x C 2 K and i n .x/ 1 C 0 for each x 2 K and n n0. n By adapting an idea first developed in [AlAm05](seealso[AlRa99]), in order to study the solutions to the diffusion problem governed by ZT we introduced and studied a further modification of the Bn’s, in symbols Mn, defined by setting

Mn.f / WD Bn ..1 C =n/ .f ı .id C ˇ=n/// for every f 2 C.K/, where id.y/ D y for every y 2 K (see [AlEtAl14, Section 5.2]). 2 Also in this case we showed that then .ZT ; C .K// is a pre-generator of a Markov semigroup for which the sequence .Mn/n1 is a strongly admissible sequence. In all the above-mentioned cases, by means of the relevant strongly admissible approximating sequences, we carried out a careful qualitative analysis of the preservation properties of the semigroups.

2.4 Final Remarks

1. Similar problems can be considered for convex compact subsets K of a (not necessarily finite dimensional) locally convex Hausdorff space X. 2. The crucial assumption (2.2) that the Markov operator T maps polynomials into polynomials of the same degree seems to have an independent interest on its own. We discuss several situations where it is satisfied in the final part of Section 4.5 of [AlEtAl14]aswellasin[AlEtAl14b]. 18 F. Altomare et al.

3. In some special cases we describe the asymptotic behavior of the Markov semigroup, i.e. we determine lim T.t/, and we characterize the saturation class t!C1 of .Bn/n1 and the Favard class of .T.t//t0 (see Subsection 4.5.4 and Appendix A.2 of [AlEtAl14]). 4. Fields of application of formula (2.6) include both probabilistic and numerical applications.

References

[AlEtAl07] Albanese, A., Campiti, M., Mangino, E.M.: Regularity properties of semigroup generated by some Fleming-Viot type operators. J. Math. Anal. Appl. 335(2), 1259–1273 (2007) [AlAm05] Altomare, F., Amiar, R.: Approximation by positive operators of the C0-semigroups associated with one-dimensional diffusion equations. I. Numer. Funct. Anal. Optim. 26, 1–15 (2005) [AlCa94] Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applica- tions. de Gruyter Studies in Mathematics, vol. 17. Walter de Gruyter, Berlin/New York (1994) [AlLe09] Altomare, F., Leonessa, V.: An invitation to the study of evolution equations by means of positive linear operators. Lect. Notes of Semin. Interdiscip. Mat. 8, 1–41 (2009) [AlRa99] Altomare, F., Ra¸sa, I.: Feller semigroups, Bernstein type operators and generalized convexity associated with positive projections. In: New Developments in Approxi- mation Theory (Dortmund, 1998). International Series of Numerical Mathematics, vol. 132, pp. 9–32. Birkhäuser, Basel (1999) [AlEtAl09] Altomare, F., Leonessa, V., Milella, S.: Cores for second-order differental operators on real intervals. Commun. Appl. Anal. 13, 353–379 (2009) [AlEtAl14] Altomare, F., Cappelletti Montano, M., Leonessa, V., Ra¸sa, I.: Markov Operators, Positive Semigroups and Approximation Processes. de Gruyter Studies in Mathe- matics, vol. 61. Walter de Gruyter GmbH, Berlin/Boston (2014) [AlEtAl14a] Altomare, F., Cappelletti Montano, M., Leonessa, V., Ra¸sa, I.: On differential operators associated with Markov operators. J. Funct. Anal. 266(6), 3612–3631 (2014) [AlEtAl14b] Altomare, F., Cappelletti Montano, M., Leonessa, V., Ra¸sa, I.: On Markov operators preserving polynomials. J. Math. Anal. Appl. 415(1), 477–495 (2014) [AlEtAl16a] Altomare, F., Cappelletti Montano, M., Leonessa, V., Ra¸sa, I.: A generalization of Kantorovich operators for convex compact subsets. Banach J. Math. Anal., advance publication, 6 May 2017. doi:10.1215/17358787-2017-0008. http://projecteuclid. org/euclid.bjma/1494036023 [AlEtAl16b] Altomare, F., Cappelletti Montano, M., Leonessa, V., Ra¸sa, I.: On the limit semigroups associated with generalized Kantorovich operators. Submitted. [AtCa00] Attalienti, A., Campiti, M.: Denerate evolution problems and beta-type operators. Stud. Math. 140, 117–139 (2000). [CaEtAl99] Campiti, M., Metafune, G., Pallara, D.: General Voronvskaja formula and solutions of second-order degenerate differential equations. Rev. Roum. Math. Pures Appl. 44(5–6), 755–766 (1999) [CeCl01] Cerrai, C., Clément, Ph.: On a class of degenerate elliptic operators arising from the Fleming-Viot processes. J. Evol. Equ. 1, 243–276 (2001) 2 Approximation Processes Generated by Markov Operators 19

[EnNa00] Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York (2000) [Ma02] Mangino, E.M.: Differential operators with second-order degeneracy and positive approximation processes. Constr. Approx. 18(3), 443–466 (2002) [MuRh11] Mugnolo, D., Rhandi, A.: On the domain of a Fleming-Viot type operator on a Lp-space with invariant measure. Note Mat. 31(1), 139–148 (2011) [Ta14] Taira K.: Semigroups, Boundary Value Problems and Markov Processes. Springer Monographs in Mathematics, 2nd edn. Springer, Heidelberg (2014) Chapter 3 Analysis of Boundary-Domain Integral Equations for Variable-Coefﬁcient Neumann BVP in 2D

T.G. Ayele, T.T. Dufera, and S.E. Mikhailov

3.1 Preliminaries

Let ˝ be a domain in 2 bounded by a smooth curve @˝, and let n.x/ be the exterior unit normal vector defined on @˝. The set of all infinitely differentiable function on ˝ with compact support is denoted by D.˝/. The function space D0.˝/ consists of all continuous linear functionals over D.˝/: The space Hs.2/, s 2 , denotes the Bessel potential space, and Hs.2/ is the dual space to Hs.2/. We define s 0 s 2 1 H .˝/ Dfu 2 D .˝/ W u D Uj˝ for some U 2 H . /g. Note that H .˝/ 1 es coincides with the Sobolev space W2 .˝/, with equivalent norms. The space H .˝/ D.˝/ s.2/ . 1 ; 1 /; is the closure of with respect to the norm of H , and for s 2 2 2 the space Hs.˝/ can be identified with Hes.˝/; see, e.g., [McL00]. We shall consider the scalar elliptic differential equation

2 Ä X @ @u.x/ Au.x/ D a.x/ D f .x/ in ˝; @x @x iD1 i i where u is unknown function and f is a given function in ˝. We assume that a.x/ 2 C1.2/; a.x/>c >0.

T.G. Ayele () Addis Ababa University, Addis Ababa, Ethiopia e-mail: [email protected] T.T. Dufera Adama Science and Technology University, Adama, Ethiopia e-mail: [email protected] S.E. Mikhailov Brunel University West London, Uxbridge, UB8 3PH, UK e-mail: [email protected]

1 From the well-known theorem of Gauss and Ostrogradski, if h 2 C0.˝/, then Z Z @ C h.x/dx D h.x/ni.x/dsx;.i D 1; 2/ (3.1) @xi ˝ @˝ where Ch.x/ is the interior boundary trace of h.x/. By the trace theorem (see, e.g., [McL00, Theorem 3.29, 3.38]), the integral relation (3.1) holds for any h 2 H1.˝/. @ . / For u 2 H2.˝/ and v 2 H1.˝/ if we put h.x/ D a.x/ u x v.x/ and applying the @xj Gauss-Ostrogradski Theorem, we obtain the following Green’s ﬁrst identity: Z Z C C E.u;v/D .Au/.x/v.x/dx C T u.x/ v.x/dsx; (3.2) ˝ @˝

2 Z X @u.x/ @v.x/ where E.u;v/WD a.x/ dx is the symmetric bilinear form, and @xi @xi iD1 ˝ Ä X2 @ TCu.x/ WD n .x/ C a.x/ u.x/ for x 2 @˝ (3.3) i @x iD1 i is the interior co-normal derivative. Thus holds the following Lemma. 2 1 C C Lemma 1 For u 2 H .˝/ and v 2 H .˝/, E.u;v/D.Au;v/˝ C.T u; v/@˝ : Remark 1 For v 2 D.˝/, Cv D 0.Ifu 2 H1.˝/, then we can deﬁne Au as a distribution on ˝ by, .Au;v/DE.u;v/for v 2 D.˝/: The subspace H1;0.˝I A/ is deﬁned as in [Cos88](see also, [Mik11])

1;0 1 H .˝I A/ WD fg 2 H .˝/ W Ag 2 L2.˝/g;

2 2 2 g 1;0 g 1 Ag : with the norm k kH .˝IA/ WD k kH .˝/ Ck kL2.˝/ For u 2 H1.˝/ the classical co-normal derivative (3.3) is not well defined, but for u 2 H1;0.˝I A/, there exists the following continuous extension of this definition hinted by the first Green identity (3.2) (see, e.g.,[Cos88, Mik11] and the references therein). Definition 1 For u 2 H1;0.˝I A/ the (canonical) co-normal derivative TCu 2 1 H 2 .@˝/ is defined in the following weak form: Z 1 C ; E. ; C / . / C 2 .@˝/ hT u wi˝ WD u 1w C Au 1wdx for all w 2 H (3.4) ˝

1 1 C 2 .@˝/ .˝/ where 1 W H ! H is a continuous right inverse of the interior C 1 1 trace operator , which maps H .˝/ ! H 2 .@˝/, while h; i@˝ denote the 3 Boundary-Domain Integral Equations in 2D 23

1 1 duality brackets between the spaces H 2 .@˝/ and H 2 .@˝/, which extend the usual L2.@˝/ inner product. Remark 2 The ﬁrst Green identity (3.2) also holds for u 2 H1;0.˝I A/ and v 2 H1.˝/ ([Cos88, Mik11]). By interchanging the role of u and v in the ﬁrst Green identity and subtracting the result, we obtain the Green second identity for u;v 2 H1;0.˝I A/, Z C C C C .vAu uAv/dx DhT u; vi@˝ hT v; ui@˝ : (3.5) ˝

We will consider the following Neumann boundary value problem: for 0 2 1 1 H 2 .@˝/, and f 2 L2.˝/ ﬁnd a function u 2 H .˝/ satisfying,

Au D f in ˝; (3.6) C T u D 0 on @˝: (3.7)

Here equation (3.6) is understood in distributional sense as in Remark 1, and equation (3.7) is understood in functional sense (3.4). The following assertion is well known, cf. e.g. [Ste08, Theorem 4.9]. Theorem 1 The homogeneous problem corresponding to the BVP (3.6)–(3.7), admits solutions in H1.˝/ spanned by u0 D 1. The nonhomogeneous problem (3.6)– (3.7) is solvable if and only if

0 C 0 hf ; u i˝ h 0; u i@˝ D 0: (3.8)

3.2 Parametrix-Based Potential Operators

Deﬁnition 2 A function P.x; y/ is a parametrix (Levi function) for the operator A if

AxP.x; y/ D ı.x y/ C R.x; y/ where ı is the Dirac-delta distribution, while R.x; y/ is a remainder possessing at most a weak singularity at x D y. The parametrix and the corresponding remainder can be chosen as in [Mik02],

2 log jx yj X x y @a.x/ P.x; y/ D ; R.x; y/ D i i ; x; y 2 2: 2 a.y/ 2 a.y/jx yj2 @x iD1 i

Similar to [Mik02, CMN09a], we deﬁne the parametrix-based Newtonian and remained potential operators as 24 T.G. Ayele et al. Z Z Pg.y/ WD P.x; y/g.x/dx; Rg.y/ WD R.x; y/g.x/dx: ˝ ˝

The single and double layer potential operators corresponding to the parametrix P.x; y/ are deﬁned for y … @˝ as Z Z . / . ; / . / ; . / C . ; / . / ; Vg y WD P x y g x dsx Wg y WD Tx P x y g x dsx @˝ @˝ where g is some scalar density function. The following boundary integral (pseudodifferential) operators are also deﬁned for y 2 @˝; Z Z V . / . ; / . / ; W . / C . ; / . / ; g y WD P x y g x dsx g y WD Tx P x y g x dsx @˝ @˝ Z W0 . / C . ; / . / ; LC . / C . /: g y WD Ty P x y g x dsx g y WD Ty Wg y @˝

C Let V; W; V; W; L be the potentials corresponding to the Laplace operator A D . Then the following relations hold (cf.[CMN09a] for 3D case), 1 1 Vg D Vg; Wg D W.ag/ (3.9) a a 1 1 Vg D Vg; Wg D W.ag/; (3.10) a a Ä Â Ã @ 1 0 0 W g D W g C a Vg; (3.11) @n a Ä Â Ã @ 1 LCg D LC.ag/ C a WC.ag/: (3.12) @n a

If u 2 H1;0.˝I A/, then substituting v.x/ by P.x; y/ in the second Green identity (3.5)for˝ n B.y;"/, where B.y;"/is a disc of radius " centered at y, and taking the limit " ! 0, we arrive at the following parametrix-based third Green identity (cf. e.g. [Mir70, Mik02, CMN09a]),

u C Ru VTCu C W Cu D PAu in ˝: (3.13)

Applying the trace operator to equation (3.13) and using the jump relation (see, e.g., [McL00, Theorem 6.11]), we have 1 Cu CRu VTCu W Cu CPAu @˝: 2 C C D on (3.14) 3 Boundary-Domain Integral Equations in 2D 25

Similarly, applying co-normal derivative operator to equation (3.13), and using again the jump relation, we obtain 1 TCu TCRu W0TCu LC Cu TCPAu @˝: 2 C C D on (3.15)

For some functions f ; and ˚ let us consider a more general indirect integral relation associated with equation (3.13),

u C Ru V C W˚ D Pf in ˝: (3.16)

1 1 1 Lemma 2 Let u 2 H .˝/; f 2 L2.˝/; 2 H 2 .@˝/; ˚ 2 H 2 .@˝/ satisfy equation (3.16). Then u belongs to H1;0.˝I A/ and is a solution of PDE (3.6), i.e., Au D fin˝ and V. TCu/.y/ W.˚ Cu/.y/ D 0; y 2 ˝. Proof The proof follows in the similar way as in the corresponding proof in 3D case in [CMN09a, Lemma 4.1]. ut s .@˝/ s.@˝/ ;1 0 Let us deﬁne the subspaces H Dfg 2 H Whg i@˝ D g (see, e.g., [Ste08, p. 147]). The following result is proved in [DM15, Lemma 2].

1 1 2 2 Lemma 3 (i) Let either 2 H .@˝/ and diam.˝/ < 1,or 2 H .@˝/. If V .y/ D 0,in˝, then D 0 on @˝. 1 (ii) Let ˚ 2 H 2 .@˝/.IfW˚ .y/ D 0,in˝, then ˚ D 0 on @˝. C C Let L denote the operator L for the constant-coefﬁcient case a Á 1. Its null- 1 space in H 2 .@˝/ includes non-zero functions. One can see this by taking u.y/ Á 1 in ˝ in the trace of the third Green identity (3.15) for the case a Á 1. Let us LO LC @a . 1 W/ LC. / @˝: introduce the operator, g WD C @n 2 I C g D ag on Theorem 2 Let @˝ be an inﬁnitely smooth boundary curve. (i) The pseudo-differential operator

1 1 2 2 LO W H.@˝/ ! H .@˝/; (3.17)

is invertible. (ii) The operator

1 1 LC LO W H 2 .@˝/ ! H 2 .@˝/; (3.18)

is bounded and compact. 1 Proof For g 2 H 2 .@˝/ using the jump relation, one can obtain the relation, LCg D L . / @a . 1 W/ LO L . / LC @a . 1 W/ ag @n 2 I C g,or g D ag D g C @n 2 I C g.The 1 1 hypersingular boundary integral operator L W H 2 .@˝/ ! H 2 .@˝/ is bounded 1 (see [DM15, Theorem 1 for s D 2 ] and the references therein). Moreover, it is 1 2 H.@˝/-elliptic (cf. [Ste08, Eq. (6.38)]). Then the Lax-Milgram lemma implies 26 T.G. Ayele et al.

1 2 the H.@˝/invertibility of L. Hence the invertibility of (3.17) follows. The 1 3 operator W W H 2 .@˝/ ! H 2 .@˝/ is continuous (see, e.g., [DM15, Theorem 1, 1 3 1 2 .@˝/ 2 .@˝/ for s D 2 ]), and since H is continuously embedded in H ,usingthe LC LO @a . 1 W/; LC LO relation D@n 2 I we obtain continuity of the operator W 1 1 1 1 H 2 .@˝/ ! H 2 .@˝/. The embedding H 2 .@˝/ H 2 .@˝/ is compact, which 1 1 implies that the operator LC LO W H 2 .@˝/ ! H 2 .@˝/ is compact. ut 1 1 Corollary 1 The operator LC W H 2 .@˝/ ! H 2 .@˝/ is Fredholm operator of index zero. 1 1 Proof The operator L W H 2 .@˝/ ! H 2 .@˝/ is Fredholm of index zero (see, 1 1 e.g.,[McL00, Theorem 7.8]). Thus, the operator LO W H 2 .@˝/ ! H 2 .@˝/ is Fredholm of index zero. Since LC D .LC LO/ C LO, it is the sum of a compact operator and a Fredholm operator of index zero and hence is also a Fredholm operator of index zero (cf. eg. [McL00, Theorem 2.26]). ut

3.3 BDIEs for Neumann BVP

To reduce the variable-coefﬁcient Neumann BVP (3.6)–(3.7)toasegregated boundary-domain integral equation system, let us denote the unknown trace by 1 ' WD Cu 2 H 2 .@˝/ and further consider ' as formally independent of u. Assuming that the function u satisﬁes the PDE Au D f , by substituting the Neumann condition into the third Green identity (3.13) and either into its trace (3.14) or into its co-normal derivative (3.15)on@˝, we can reduce the BVP (3.6)–(3.7) to two different systems of Boundary-Domain Integral equations 1;0 1 for the unknown functions u 2 H .˝I A/ and ' WD Cu 2 H 2 .@˝/. BDIE system (N1), obtained from the third Green identity (3.13) and its co- normal derivative (3.15), is

u C Ru C W' D G0 in ˝; C C C T Ru C L ' D T G0 0 on @˝; where

G0 WD Pf C V 0 in ˝: (3.19)

C CP W0 1 Also note that T G0 D T f C 0 C 2 0. The system can be written in matrix 1 1 1;0 1 operator form as N U D G where U WD Œu;'t 2 H .˝I A/ H 2 .@˝/ and Ä Ä R N1 I C W ; G1 G0 : D C C D C (3.20) T RL T G0 0 3 Boundary-Domain Integral Equations in 2D 27

From the mapping properties of the operators V and P in [DM15, Theorems 1;0 1 and 3], we get the inclusion G0 2 H .˝I A/, and Deﬁnition 1 implies C 1 1 1 1 T G0 2 H 2 .@˝/. Therefore, G belongs to H .˝/ H 2 .@˝/. Due to the mapping properties of the operators involved in N1, the operator N1 W H1;0.˝I A/ 1 1 1 H 2 .@˝/ ! H .˝/ H 2 .@˝/ is bounded. BDIE system (N2), obtained from the third Green identity (3.13) and its trace (3.14), is

u C Ru C W' D G0 in ˝; 1 C C Ru ' W' G0 @˝; C 2 C D on

2 where G0 is given by the relation (3.19). In a matrix form it can be written as N U D G2 where Ä Ä 2 I C R W 2 G0 N D ; G D : (3.21) CR 1 W C 2 I C G0

C 1 2 1 1 By the trace theorem G0 2 H 2 .@˝/. Therefore, G 2 H .˝/ H 2 .@˝/.The mapping properties of operators involved in N2 imply the operator N2 W H1;0.˝I A/ 1 1 1 H 2 .@˝/ ! H .˝/ H 2 .@˝/ is bounded.

1 1 2 2 Remark 3 Let 0 2 H .@˝/ and diam.˝/ < 1,or 0 2 H .@˝/. Then 2 G D 0 if and only if .f ; 0/ D 0: Indeed, the latter equality evidently implies 2 C the former. Inversely, if G D 0, then G0 D 0 and G0 D 0. Then, G0 D 0 implies Pf C V 0 D 0 in ˝. Multiplying by a, taking into consideration that aV D V is harmonic and applying Laplace operator, we get f D 0. And hence V 0 D 0 in ˝. Then by Lemma 3(i), 0 D 0 on @˝.

3.4 Equivalence and Invertibility Theorems

In the following theorem we shall see the equivalence of the original boundary value problem with the boundary-domain integral equation systems.

1 Theorem 3 Let 0 2 H 2 .@˝/ and f 2 L2.˝/ satisfy the solvability condition (3.8). (i) If some u 2 H1.˝/ solves the Neumann BVP (3.6)–(3.7), then the pair .u;'/, where

1 ' D Cu 2 H 2 .@˝/; (3.22)

solves the BDIE systems (N1) and (N2). 28 T.G. Ayele et al.

1 1 (ii) If a pair .u;'/2 H .˝/H 2 .@˝/ solves the BDIE systems (N1), then u solves BDIE system (N2) and Neumann BVP (3.6)–(3.7), and ' satisfies (3.22). 1 1 (iii) If a pair .u;'/ 2 H .˝/ H 2 .@˝/ solves the BDIE systems (N2), and diam.˝/ < 1, then u solves BDIE system (N1) and Neumann BVP (3.6)–(3.7), and ' satisfies (3.22). (iv) The homogeneous BDIE systems (N1) and (N2) have linearly independent 0 0 0 1 1 solutions spanned by U D .u ;' /T D .1; 1/T in H .˝/ H 2 .@˝/. Condition (3.8) is necessary and sufficient for solvability of the nonhomogeneous BDIE system (N1) and, if diam.˝/ < 1, also of the system (N2), in 1 1 H .˝/ H 2 .@˝/. Proof (i) Let u 2 H1.˝/ be solution of the Neumann BVP (3.6)–(3.7). Since 1;0 C f 2 L2.˝/, then u 2 H .˝I A/. Setting ' D u, and recalling how BDIE systems (N1) and (N2) were constructed, we obtain that .u;'/solves them. 1 1 (ii) Let now a pair .u;'/ 2 H .˝/ H 2 .@˝/ solve the system (N1) or (N2). Due to the first equations in the BDIE systems, the hypotheses of Lemma 2 are satisfied implying that u belongs to H1;0.˝I A/ and solves PDE (3.6)in˝, while the following equation also holds:

C C V. 0 T u/.y/ W.' u/.y/ D 0; y 2 ˝: (3.23)

1 1 If a pair .u;'/ 2 H .˝/ H 2 .@˝/ solve the system (N1) then taking the co-normal derivatives of the first equation in (N1) and subtracting the second C from it, we get T u D 0 on @˝. Thus the Neumann condition is satisfied, and using it in (3.23) we get W.' Cu/.y/ D 0 on y 2 @˝. Lemma 3(ii) implies ' D Cu on @˝. 1 1 (iii) Let now a pair .u;'/ 2 H .˝/ H 2 .@˝/ solves the system (N2). Taking the trace of the first equation in (N2) and subtracting the second from it, we get C C ' D u on @˝. Then inserting it in (3.23)givesV. 0 T u/.y/ D 0 on y 2 C @˝. Lemma 3(i) implies 0 D T u on @˝. Hence the Neumann condition is satisfied. (iv) Theorem 1 along with item (i)–(iii) implies the claims of item (iv). ut Note that Theorem 3(iv) implies that the operators

1 1 1 1 1 N W H .˝/ H 2 .@˝/ ! H .˝/ H 2 .@˝/; (3.24) and

2 1 1 1 1 N W H .˝/ H 2 .@˝/ ! H .˝/ H 2 .@˝/; (3.25) are not injective. 3 Boundary-Domain Integral Equations in 2D 29

Theorem 4 Operators (3.24) and (3.25) are Fredholm operator with zero index. Moreover, the kernels (null-spaces) of these operator are spanned by the element .u0;'0/ D .1; 1/ and thus the kernels and co-kernels of the operators are one- dimensional. 1 1 Proof (i) By Corollary 1, the operator LO W H 2 .@˝/ ! H 2 .@˝/ is a Fredholm operator with zero index. Therefore, the operator Ä 1 IW 1 1 1 1 N .˝/ 2 .@˝/ .˝/ 2 .@˝/; 0 WD 0 LO W H H ! H H

is also Fredholm with zero index. By the properties of operators R and TCR (see, e.g., [DM15, Corollary 2], and the reference therein) and Theorem 2(ii), the operator Ä 1 1 R 0 1 1 1 1 N N D W H .˝/ H 2 .@˝/ ! H .˝/ H 2 .@˝/; 0 TCRLC LO

is compact, implying that operator (3.24) is Fredholm with zero index (see [McL00, Theorem 2.26]). (ii) Let us consider the operator Ä N2 IW 0 D 0 1 2 I

2 1 1 1 1 Then, N0 W H .˝/ H 2 .@˝/ ! H .˝/ H 2 .@˝/ is bounded. It is invertible due to its triangular structure and invertibility of its diagonal operators, I W 1 1 1 1 H .˝/ ! H .˝/; and I W H 2 .@˝/ ! H 2 .@˝/. Due to the compactness properties of R; CR, and W (see, e.g., [DM15, Corollary 1 and 2], the operator Ä 2 2 R 0 1 1 1 1 N N D W H .˝/ H 2 .@˝/ ! H .˝/ H 2 .@˝/; 0 CRW

is compact. This implies that the operator (3.25) is a Fredholm operator with zero index. (iii) The kernels of the operators are spanned by the element .u0;'0/ D .1; 1/ due to Theorem 3(iv). ut The following theorem describes the co-kernels of these operators. The proof is similar to the proofs of the corresponding assertions for 3D case in [Mik15]. 3 s 2 s2 Let W H .@˝/ ! H@˝ denote the operator adjoint to the trace operator W 2 2 3 3 s. / 2 s.@˝/ < H ! H ,fors 2 . 30 T.G. Ayele et al.

Theorem 5 Let diam.˝/ < 1 and u0.x/ D 1. (i) The co-kernel of operator (3.24) is spanned over the functional g1 2 He1.˝/ 1 H 2 .@˝/ deﬁned as ! C 1 C 0 a V u g1 WD : (3.26) 0

(ii) The co-kernel of operator (3.25) is spanned over the functional g2 2 He1.˝/ 1 H 2 .@˝/ deﬁned as ! C. 1 W0 /V1 C 0 a 2 C u g2 D : (3.27) . 1 W0 /V1 C 0 a 2 u

3.5 Perturbed BDIE Systems for the Neumann Problem

Theorem 3 implies that even when the solvability condition (3.8) is satisfied, the solutions of both BDIE systems, (N1) and (N2), are not unique. By Theorem 4,in turn, the BDIE left-hand side operators, N1 and N2, have non-zero kernels and thus are not invertible. To find a solution .u;'/ from uniquely solvable BDIE systems with continuously invertible left-hand side operators, let us consider, following [Mik99], some BDIE systems obtained form (N1) and (N2) by finite-dimensional operator perturbations, cf.[Mik15] forR the three-dimensional case. Below we use the T notations U D .u;'/ and j@˝jWD @˝ ds: Perturbation of BDIE system (N1): Let us introduce the perturbed counterparts of the BDIE system (N1),

NO 1U D G1; (3.28) Z Â Ã 1 1 1 1 1 0 1 1 a .y/ NO WD N C NV and NV U.y/ WD g .U/Z .y/ D '.x/ds ; j@˝j j@˝j 0 that is, Z Â Ã 1 0 0 1 1 a .y/u .y/ g .U/ WD '.x/ds; Z .y/ WD : j@˝j j@˝j 0

1 1 For the functional g given by (3.26) in Theorem 5, since the operator V W 1 1 H 2 .@˝/ ! H 2 .@˝/ is positive deﬁnite (with additional condition diam.˝/ < 1) and u0.x/ D 1, there exists a positive constant C such that 3 Boundary-Domain Integral Equations in 2D 31

1 1 C 1 C 0 1 0 1 C 0 C 0 g .Z / Dha V u ; a u i˝ DhV u ; u i@˝ C 0 2 C 0 2 @˝ 2 <0: ÄCk u k 1 ÄCk u kL2.@˝/ DCj j (3.29) H 2 .@˝/

0 0 0 1 1 Further, for U D .u ;' /T D .1; 1/T in H .˝/ H 2 .@˝/, Z 1 g0.U0/ D Cu0ds D 1: (3.30) j@˝j @˝

Due to (3.29) and (3.30), [Mik99, Lemma 2] implies the following assertion: Theorem 6 Let diam.˝/ < 1, then 1 1 1 1 1 (i) The operator NO W H .˝/ H 2 .@˝/ ! H .˝/ H 2 .@˝/ is continuous and continuously invertible. (ii) If g1.G1/ D 0 (or condition (3.8)forG1 in form (3.20) is satisﬁed), then the unique solution of perturbed BDIDE system (3.28) gives a solution of original BDIE system (N1) such that Z 1 g0.U/ D 'ds D 0: j@˝j @˝

Perturbation of BDIE system (N2): Let us introduce the perturbed counterparts of the BDIE system (N2)

NO 2U D G2; (3.31) where Z Â Ã 1 1. / NO 2 N2 NV 2 NV 2U. / 0.U/F 2. / '. / a y ; WD C and y WD g y D x ds C 1 j@˝j j@˝j a .y/ that is, Z Â Ã 1 1. / 0. / 0.U/ '. / ; F 2. / a y u y : g WD x ds y WD C 1 0 j@˝j j@˝j Œa u .y/

2 1 For the functional g given by (3.27) in Theorem 5(ii), since the operator V W 1 1 H 2 .@˝/ ! H 2 .@˝/ is positive deﬁnite (with additional condition diam.˝/ < 1) and u0.x/ D 1, there exists a positive constant C such that

1 2 2 C 0 1 C 0 1 0 g .F / a . W /V u ; a u ˝ Dh 2 C i 1 0 1 C 0 C 1 0 a. W /V u ; .a u / @˝ Ch 2 i 32 T.G. Ayele et al.

1 1 0 1 C 0 0 1 C 0 C 0 1 C 0 C 0 . W /V u . W /V u ; u @˝ V u ; u @˝ Dh 2C C 2 i Dh i C 0 2 C 0 2 @˝ 2 <0: ÄCk u k 1 ÄCk u kL2.@˝/ DCj j (3.32) H 2 .@˝/

Due to (3.32) and (3.30), [Mik99, Lemma 2] implies the following assertion. Theorem 7 Let diam.˝/ < 1, then 2 1 1 1 1 (i) The operator NO W H .˝/ H 2 .@˝/ ! H .˝/ H 2 .@˝/ is continuous and continuously invertible. (ii) If g2.G2/ D 0 (or condition (3.8)forG2 in form (3.21) is satisﬁed), then the unique solution of perturbed BDIDE system (3.31) gives a solution of original BDIE system (N2) such that Z Z 1 1 g0.U/ D Cuds D 'ds D 0: j@˝j @˝ j@˝j @˝

3.6 Conclusion

In this paper, we have considered the interior Neumann boundary value problem for a variable-coefﬁcient PDE in a 2D domain, where the right-hand side function is 1 from L2.˝/ and the Neumann data from the space H 2 .@˝/. The BVP was reduced to two systems of Boundary-Domain Integral Equations and their equivalence to the original BVP was shown. The null-spaces of the corresponding BDIE systems are not trivial. Moreover, the BDIE systems are neither uniquely nor unconditionally solvable. The BDIE operators for the Neumann BVP are bounded but only Fredholm with zero index. The kernels and co-kernels of these operators were analyzed, and appropriate ﬁnite-dimensional perturbations were constructed to make the perturbed operators invertible and provide a solution of the original BDIE systems and of the Neumann BVP. In a similar way one can consider also the 2D versions of the BDIEs for other BVP problems in interior and exterior domains, united BDIEs as well as the localized BDIEs, which were analyzed for 3D case in [CMN09a, CMN13, Mik06, CMN09b].

Acknowledgements The work on this paper of the ﬁrst author was supported by the Alexander von Humboldt Foundation Return Fellowship Grant No. 3.4-RKS-ATH/1144171, and of the third author by the grant EP/M013545/1: “Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs” from the EPSRC, UK. 3 Boundary-Domain Integral Equations in 2D 33

References

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E. Berchio, D. Buoso, and F. Gazzola

4.1 Introduction

From the Federal Report [AmEtAl41] on the Tacoma Narrows Bridge collapse (see also [Sc01]), we learn that the most dangerous oscillations for the deck of a bridge are the torsional ones, which appear when the deck rotates around its main axis; we refer to [Ga15, §1.3,1.4] for a historical survey of several further collapses due to torsional oscillations. These events naturally raise the following question: is it possible to measure the torsional performances of bridges? Following [FeEtAl15] we model the deck of a bridge as a long narrow rectangular thin plate ˝ hinged at two opposite edges and free on the remaining two edges; this well describes the deck of a suspension bridge which, at the short edges, is supported by the ground. Then we introduce a new functional, named gap function, able to measure the torsional performances of the bridge. Roughly speaking, this functional measures the gap between the displacements of the two free edges of the deck, thereby giving a measure of the risk for the bridge to collapse. In the present paper we explicitly compute the gap function for some prototypes of external forces that appear to be the most prone to generate torsional instability in the structure. Our theoretical and numerical results conﬁrm that the gap function is a reliable measure for the torsional performances of rectangular plates.

E. Berchio () • D. Buoso Politecnico di Torino, Torino, Italy e-mail: [email protected]; [email protected] F. Gazzola Polytechnic University of Milano, Milano, Italy e-mail: ﬁ[email protected]

4.2 Variational Setting and Gap Function Deﬁnition

Up to scaling, we may assume that ˝ D .0; / .`;`/ 2 with 2` . According to the Kirchhoff-Love theory [Ki50, Lo27], the energy ¾ of the vertical deformation u of the plate ˝ may be computed by Z Â Ã 1 ¾. / . /2 .1 /. 2 / ; u D u C uxy uxxuyy fu dx dy ˝ 2 where 0<<1n is the Poisson ratio. The functionalo ¾ is minimized on the space 2 2.˝/ 0 0; . `;`/ ˝ 2 2.˝/ 0.˝/ H WD w 2 H I w D on f g I since , H C so that the condition on f0; g.`;`/ is satisﬁed pointwise. By [FeEtAl15, 2 Lemma 4.1], H is a Hilbert space when endowed with the scalar product R . ;v/ 2 v .1 /.2 v v v / u H WD ˝ u C uxy xy uxx yy uyy xx dxdy

2 and associated norm kuk 2 D .u; u/ 2 , which is equivalent to the usual norm H H 2.˝/ 2 2 ; in H . We also deﬁne H the dual spaceR of H and we denote by h i the 2 ; corresponding duality. If f 2 H we replace ˝ fu with hf ui. In such case, there 2 exists a unique u 2 H such that

2 . ;v/ 2 ;v v ; u H Dhf i82 H (4.1) and u is the minimum point of the functional ¾.Iff 2 L2.˝/, then u 2 H4.˝/ and u is a strong solution of the problem 8 ˆ 2 <ˆ u D f in ˝ .0; / .0; / . ; / . ; / 0 . `;`/ ˆu y D uxx y D u y D uxx y D for y 2 :ˆ uyy.x; ˙`/ C uxx.x; ˙`/ D uyyy.x; ˙`/ C .2 /uxxy.x; ˙`/ D 0 for x 2 .0; /: (4.2)

See [FeEtAl15, Ma05] for the derivation of (4.2). Differently from what happens in 2 higher space dimension or for lower order problems, here u 2 H is continuous if f 1 2 merely belongs to L or H and we can deﬁne the gap function:

G.x/ WD u.x;`/ u.x; `/ x 2 .0; /: (4.3)

G measures the difference of the vertical displacements on the free edges and is therefore a measure of the torsional response. The maximal gap is given by ˇ ˇ G1 WD max ˇu.x;`/ u.x; `/ˇ : (4.4) x2Œ0;

This gives a measure of the risk for the bridge to collapse. Clearly, G1 depends on f and our purpose is to compute the torsional performances of ˝ for different f . 4 Torsional Performances of Rectangular Plates 37

When f D f .x/ the explicit solution of (4.2) was obtained in [FeEtAl15]by adapting the separation of variables approach of [Ma05, Section 2.2]. In fact, a similar procedure can be used also for some forcing terms depending on y such as e˛yg.x/ or yg.x/,see[Ma05, p.41]. Here we focus our attention on loads of the form ˛ ˛y 2 f D f˛.x; y/ D K˛e g.x/; g 2 L .0; / K˛ WD R : 2 sinh.˛`/ 0 jg.x/j dx (4.5) 1 Hence, kf˛kL1 D . Furthermore, we write Z C1X 2 g.x/ D m sin.mx/; m D g.x/ sin.mx/ dx : (4.6) 0 mD1

We will show in Lemma 1 that f˛, with ˛ big, is a good approximation of a load concentrated on a long edge of the plate. This is a good model for the restoring force due to the hangers in a suspension bridge because they act close to the free edges. We prove Theorem 1 Assume that f satisﬁes (4.5)–(4.6) for some ˛ > 0, ˛ … . Then the unique solution of (4.2) is given by

C1 Ä X ˛y m e u.x; y/ D K˛ C A cosh.my/ C B sinh.my/ C Cy cosh.my/ C Dy sinh.my/ sin.mx/ .m2 ˛2/2 mD1 where the constants A D A.m;`;˛/,B D B.m;`;˛/,C D C.m;`;˛/, D D D.m;`;˛/are the solutions of the systems (4.17) provided in Section 4.3. By Theorem 1, the gap function (4.3) can be computed for all f satisfying (4.5)– (4.6). Here, for the sake of simplicity, we focus on the case where the concentration occurs at the midpoint of the upper edge of the plate. Namely,

˛ e˛y sin x f D f˛.x; y/ WD >0in ˝ (4.7) 4 sinh.˛`/

1 so that kf˛kL1 D . The unique solution u D u˛ of (4.2) with f D f˛ as in (4.7)is given by Theorem 1 with 1 D 1 and m D 0 if m ¤ 1.Weset

˛ ˛ B sinh ` ˛ C ` cosh ` E.`; ˛/ WD C C ; (4.8) 2.1 ˛2/2 2 sinh.˛`/ 2 sinh.˛`/ with B D B.1;`;˛/and C D C.1;`;˛/as in Theorem 1,see(4.18). Finally, we set

sinh2.`/ E.`/ WD (4.9) .1 /Œ.1 /`C .3 C /sinh.`/ cosh.`/ and we prove 38 E. Berchio et al.

Fig. 4.1 The plot of the map 1 ˛ 7! G˛ in logarithmic scale for D 0:2 and ` D 150

Theorem 2 Let u˛ be the unique solution of (4.2) with f D f˛ as in (4.7). Let G˛ 1 and G˛ be as in (4.3) and in (4.4) with u D u˛. Assume (4.8) and (4.9). Then,

G˛ .x/ D E.`; ˛/ sin x ! G.x/ WD E.`/ sin x uniformly on Œ0; as ˛ !C1: (4.10) In particular, Â Ã 1 1 .1 C /cosh ` sinh ` C .1 /` 1 G˛ D E.`; ˛/ D E.`/ C o as ˛ !C1: ˛ 2.1/Œ.3C/cosh.`/ sinh.`/ C .1/` ˛ (4.11)

G is the gap function of the problem (4.1) when f is a measure concentrated on the upper edge, see Lemma 1. On the other hand, Figure 4.1 supports the conjecture 1 that the map ˛ 7! G˛ is strictly increasing. Hence, if the same total load approaches the free edges, then the gap function increases: this validates G1 as a measure of the torsional performances. A physical interesting case is when f is in resonance with the structure, namely when it is a multiple of an eigenfunction of 2 under the boundary conditions in (4.2). Let us brieﬂy recall some known facts from [BeEtAl, FeEtAl15]. The eigenvalues of 2 under the boundary conditions in (4.2) may be ordered in an increasing sequence of strictly positive numbers diverging to C1. 2 Furthermore, the corresponding eigenfunctions form a complete system in H.More precisely, they are identiﬁed by two indices m; j 2 C and they have one of the following forms:

wm;j.x; y/ D vm;j.y/ sin.mx/ with corresponding eigenvalues m;j;

wm;j.x; y/ D vm;j.y/ sin.mx/ with corresponding eigenvalues m;j:

The vm;j are odd while the vm;j are even and, since jyj <`small, one qualitatively has vm;j.y/ ˛m;j y and vm;j.y/ ˇm;j for some constants ˛m;j and ˇm;j.This is why the wm;j are called torsional eigenfunctions while the wm;j are called longitudinal eigenfunctions; see [BeEtAl, BeEtAl16, FeEtAl15]. Here we consider the normalized eigenfunctions 4 Torsional Performances of Rectangular Plates 39

wm;j.x; y/ fm;j.x; y/ WD : (4.12) kwm;jkL1

We do not consider forcing terms proportional to the longitudinal eigenfunctions since they are even with respect to y and the corresponding gap functions vanish identically. When f is as in (4.12), we can determine explicitly the gap function. Theorem 3 Let `>0, 0<<1be such that the unique positive solution s >0of p p . 2 `/ 2 2 ` tanh s D 2 s (4.13) is not an integer; let m; j 2 . Let um;j be the unique solution of (4.2) with f D fm;j G G1 as in (4.12),let m;j and m;j be as in (4.3) and in (4.4) with u D um;j. There exist .`/ > 0 G . / . / G1 constants Cm;j D Cm;j such that m;j x D Cm;j sin mx , m;j D Cm;j, and

C ; .`/ C ; .`/ 0

The reason of (4.13) will become clear in Section 4.5 where we give the explicit dependence Cm;j D Cm;j.`/,see(4.26). Condition (4.13) has probability 0 to occur among all possible random choices of ` and : if it is violated, Theorem 3 still holds G1 .`/ 0 ` with a slight modiﬁcation of the proof. Even if the convergence m;j ! as ! 0 appears reasonable, a surprise arises from (4.14) which states that the convergence is at the third power: this is due to the behavior of the torsional eigenvalues. Take 0:2 ` again D and D 150 (two reasonable values for plates modeling the deck of a bridge), by using (4.26), we numerically obtained the values in Table 4.1 below. Due to the fact that w1;1 is the only torsional eigenfunction with two nodal regions, one expects from a reliable measure G1 of the torsional performances of ˝ to satisfy

G1 G1 : max m;j D 1;1 m;j2C

G1.`/ ` =150 0:2 Table 4.1 Numerical values of m;j when D and D jnm 1 2 3 4 5 1 4:3629 103 1:0904 103 4:8439 104 2:7229 104 1:7411 104 2 4:3566 108 4:3555 108 4:3536 108 4:3509 108 4:3474 108 3 4:1216 109 4:1214 109 4:1209 109 4:1203 109 4:1195 109 4 9:4439 1010 9:4436 1010 9:4432 1010 9:4426 1010 9:4418 1010 5 3:2251 1010 3:2250 1010 3:2249 1010 3:2248 1010 3:2247 1010 40 E. Berchio et al.

G1 G1 Table 4.1 supports this conjecture and shows that the maps m 7! m;j and j 7! m;j 1 are decreasing. Note that G1;1 0:0043 < E. =150/ 0:0065 which is the value in (4.11) when ˛ !C1. Nevertheless, the maximal gap of f1;1 is the worst among linear combinations of fm;j, namely when a finite number of modes is excited. C Theorem 4 Let M be a finite subset of indicesP and let mo D minfm 2 Mg. . ; / ˛ . ; / Let u be the unique solution of (4.2) with f x y D m2M mfmP;j.m/ x y , where ˛ ˛ 0 ˛ 6 1 the fm;j are defined in (4.12) and the m 2 satisfy m ¤ and m2M j mj . 6 1 . / Hence, kf kL1 . Furthermore, assume that the j D j m 2 are such that the map m 7! Cm;j.m/ is nonincreasing with the Cm;j as in Theorem 3, see also (4.26) below. Finally, let G1 be as in (4.4) with u D u; then

G1 max D Cmo;j.mo/ ˛m where the maximum is taken among all the ˛m satisfying the above restrictions. Concerning the existence of a set of indices fj.m/g such that the map m 7! Cm;j.m/ is nondecreasing, as required by Theorem 4, Table 4.1 suggests that this assumption should hold by taking j.m/ D j0 for some positive integer j0 ﬁxed.

4.3 Proof of Theorem 1

4 By linearity, we may take K˛ D 1.Let 2 H .0; / be the unique solution of 0000.x/ C 2˛2 00.x/ C ˛4 .x/ D g.x/.0

P ˛ . / C1 m . / 00 2.0; / For … ,by(4.6) we get x D mD1 .m2˛2/2 sin mx , 2 H and the corresponding series converges in H2.0; / and uniformly. By (4.15), 2Œe˛y .x/ D e˛yg.x/. Therefore, v.x; y/ WD u.x; y/e˛y .x/, with u solving (4.2), satisﬁes 8 ˆ 2v D 0 in ˝ <ˆ v D vxx D 0 on f0; g.`;`/ ˆ v v ˙˛`Œ˛2 . / 00. / .0; / ` :ˆ yy C xx De x C x on f˙ g ˙˛` 2 00 vyyy C .2 /vxxy D˛e Œ˛ .x/ C .2 / .x/ on .0; / f˙`g: (4.16) P . / v. ; / C1 . / . / We seek functions Ym y such that x y D mD1 Ym y sin mx solves (4.16). 0000. / 2 2 00. / 4 . / 0 . `;`/ . / Then Ym y m Ym y C m Ym y D for y 2 , hence Ym y D A cosh.my/ C B sinh.my/ C Cy cosh.my/ C Dy sinh.my/. By imposing the boundary conditions in (4.16), we get a 4 4 system that may be split into two 2 2 systems: 4 Torsional Performances of Rectangular Plates 41

( .1 / 2 . `/ Œ2 . `/ .1 / ` . `/ m2˛2 .˛`/ m cosh m ACm cosh m C m sinh m DD m .m2˛2/2 cosh 2 2 . 1/ 3 . `/ 2Œ.1 / . `/ . 1/ ` . `/ ˛ .2/m ˛ .˛`/; m sinh m ACm C sinh m C m cosh m DD m .m2˛2/2 sinh (4.17) ( .1 / 2 . `/ Œ2 . `/ .1 / ` . `/ m2˛2 .˛`/ m sinh m BCm sinh m C m cosh m C D m .m2˛2/2 sinh 2 2 . 1/ 3 . `/ 2Œ.1 / . `/ . 1/ ` . `/ ˛ .2/m ˛ .˛`/: m cosh m BCm C cosh m C m sinh m C D m .m2˛2/2 cosh

The proof of Theorem 1 is so complete.

4.4 Proof of Theorem 2

When f is as in (4.7)wehave m D 0 for all m > 2 and the systems (4.17) yield

A.m;`;˛/D B.m;`;˛/D C.m;`;˛/D D.m;`;˛/D 0 8m > 2 while, for m D 1, 1 D 1 and by solving (4.17) we get

.1C/.˛2/ sinh.`/ cosh.˛`/C.1/.˛2/`cosh.`/ cosh.˛`/ A D .1/.˛21/2Œ.3C/cosh.`/ sinh.`/.1/` 2˛.˛2C2/ cosh.`/ sinh.˛`/C.1/˛.˛2C2/` sinh.`/ sinh.˛`/ C .1/.˛21/2Œ.3C/cosh.`/ sinh.`/.1/`

.1C/.˛2/ cosh.`/ sinh.˛`/C.1/.˛2/`sinh.`/ sinh.˛`/ B D .1/.˛21/2Œ.3C/cosh.`/ sinh.`/C.1/` 2˛.˛2C2/ sinh.`/ cosh.˛`/C.1/˛.˛2C2/` cosh.`/ cosh.˛`/ (4.18) C .1/.˛21/2Œ.3C/cosh.`/ sinh.`/C.1/`

˛.2˛2/ sinh.`/ cosh.˛`/C.˛2/ cosh.`/ sinh.˛`/ ; C D .˛21/2Œ.3C/cosh.`/ sinh.`/C.1/`

˛.2˛2/ cosh.`/ sinh.˛`/C.˛2/ sinh.`/ cosh.˛`/ ; D D .˛21/2Œ.3C/cosh.`/ sinh.`/.1/`

and the explicit form of u˛ follows from Theorem 1. In particular, the corresponding 1 gap function G˛ is as in (4.10). Hence, G˛ D E.`; ˛/, with E.`; ˛/ is as in (4.8). 1 In order to compute the limit of G˛ as ˛ !C1, we prove ˛ . ; / sin x ı . / 2 Lemma 1 As !C1we have that f˛ x y ! 2 ` y in H that is, Z Z 1 . ; /v. ; / v. ;`/ v 2 : lim f˛ x y x y dxdy D sin x x dx 8 2 H (4.19) ˛!1 ˝ 2 0 42 E. Berchio et al.

v 2 Proof Take 2 H, integrating by parts, we have Z Z Z ! 1 ` . ; /v. ; / ˛yv. ; / ` ˛yv . ; / : f˛ x y x y dxdy D sin x e x y ` e y x y dy dx ˝ 4 sinh.˛`/ 0 ` R 1 ` ˛y ˛ v . ; / 0 By Lebesgue Theorem, lim !1 4 sinh.˛`/ ` e y x y dy D . Since, for all x 2 ` ˛yv. ; / e x y ` v.x;`/ .0; / ˛ , lim !1 4 sinh.˛`/ D 2 ,(4.19) follows. ut By letting ˛ !C1, the constants in (4.18) have the following asymptotic behavior:

2 cosh.`/ C .1 /`sinh.`/ e˛` e˛` A DW A ; .1/Œ.3C/cosh.`/ sinh.`/ .1/` 2˛ ˛ 2 sinh.`/ C .1 /`cosh.`/ e˛` e˛` B DW B ; .1/Œ.3C/cosh.`/ sinh.`/ C .1/` 2˛ ˛ sinh.`/ e˛` e˛` C DW C ; .3C/cosh.`/ sinh.`/ C .1/` 2˛ ˛ cosh.`/ e˛` e˛` D DW D : .3C/cosh.`/ sinh.`/ .1/` 2˛ ˛ . ; / . / . / . / . / sin x Set u x y WD A cosh y C B sinh y Cy cosh y Dy sinh y 2 . Clearly, one has u˛.x; y/ ! u .x; y/ a.e. in ˝ as ˛ !C1. Moreover, we show Lemma 2 As ˛ !C1we have that

. ; / . ; / 2 : u˛ x y ! u x y in H

1 sin x ı . / 2 Proof By deﬁnition, kf˛kL1 D , and by Lemma 1 f˛ ! f WD 2 ` y in H . 2 By (4.1) with f D f˛ and with f D f , there exists a (unique) uO 2 H such that 2 . ;v/ 2 ;v v ˛ 2 6 ˛ 2 : ˛. ; / uO H Dhf i for all 2 H and ku OukH kf f kH Hence, u x y ! . ; / 2 uO x y in H and, in turn, uO D u. ut The gap function corresponding to u is G.x/ D u .x;`/ u .x; `/ D E.`/ sin x ; where E.`/ is as in (4.9). By Lemma 2,as˛ !C1,wehave max jG˛ .x/G.x/j 6 max ju˛ .x;`/u .x;`/ jC max ju˛ .x; `/u .x; `/ j!0; x2Œ0; x2Œ0; x2Œ0;

1 1 and, by (4.10), G˛ ! G D E.`/. Finally, since Ä Â Ã e˛` .1 /`sinh.`/ .1 C /cosh.`/ 1 B D B C C o as ˛ !C1; ˛ 2˛.1/Œ.3C/cosh.`/ sinh.`/ C .1/` ˛ Ä Â Ã e˛` cosh.`/ 1 C D C C C o as ˛ !C1; ˛ 2˛Œ.3C/cosh.`/ sinh.`/ C .1/` ˛ we get the asymptotic in (4.11). This concludes the proof of Theorem 2. 4 Torsional Performances of Rectangular Plates 43

4.5 Proofs of Theorems 3 and 4

We state some properties of the eigenvalues and eigenfunctions of 2 by slightly improving Theorem 2.1 and Proposition 2.2 in [BeEtAl]. By (4.13), two cases may occur. If p p . 2 `/ > 2 2 `; tanh m 2 m (4.20) then m is small and the torsional eigenfunction wm;j with j > 1 is given by 2 q Á q Á 3 1=2 1=2 sinh y Cm2 sin y m2 4 1=2 2 m;j 1=2 2 m;j 5 w ; .x; y/D .1/m q Á C C.1/m q Á .mx/ m j m;j 1=2 m;j 1=2 sin ` C 2 ` 2 sinh m;j m sin m;j m (4.21) 2 where the corresponding eigenvalue m;j is the j-th solution j > m of the equation q q q q p p 2 p p p 2 p m2 C.1/m2 tanh.` Cm2/D Cm2 .1/m2 tan.` m2/I q (4.22) > 1 > 1 > 4 ` 1=2 2= For any m and j we have m;j m and m;j m … , so that the functions in (4.21) are welldeﬁned. Related to (4.22), we consider the function ( Â Ã p p ) 2 2 2 2 p 2 sC.1/m .` sCm / .` sm / . / 2 4 .1 / 2 tanh tan H s WD s m s m 2 p p s.1/m sCm2 sm2

p 2 DW s2 m4 s.1/m2 Z.s/ 2 2 1 2 with s ¤ m C 2 2 C k , k 2 . In each of the subintervals of deﬁnition for ` p p 2 sC.1/m2 tanh.` sCm2/ tan.` sm2/ Z (and s > m ), the maps s 7! 2 , s 7! p , and s 7! p s.1/m sCm2 sm2 are strictly decreasing, the ﬁrst two being positive. Since, by (4.20), lim 2 Z.s/ D p s!m 2 2 tanh. 2`m/ `>0 p , Z starts positive, ends up negative and it is strictly 2 m p decreasing in any subinterval, it admits exactly one zero there, when tan.` sm2/ is positive. Hence, H has exactly one zero on any interval and we have proved Á 2 2 . 1/2 2; 2 2 . 1 /2 2 > 1: m;j D j 2 m C `2 j m C `2 j 2 8j (4.23)

Slightly different is the second case. If p p . 2 `/ < 2 2 `; tanh m 2 m (4.24)

2 4 4 m is large, .1 /m < m;1 < m , and the ﬁrst torsional eigenfunction is 2 q Á q Á 3 1=2 1=2 sinh y Cm2 sinh y m2 4 1=2 2 m;1 1=2 2 m;1 5 w ;1.x; y/D .1/m q Á C C.1/m q Á .mx/: m m;1 1=2 m;1 1=2 sin ` C 2 ` 2 sinh m;1 m sinh m m;1 (4.25) 44 E. Berchio et al.

> 2 > 4 For j , m;j m and the eigenfunctions are still given by (4.21). Now Z has no . 4; 2 2 2/ > 2 zero in m m C 4`2 . However, for j ,(4.23) still holds. We may now deﬁne 4 C ; .`/ WD : (4.26) m j 1=2 m;j kwm;jkL1

. ; / . ; / wm;j x y The solution of (4.2) with f D fm;j is um;j x y D . Hence, from (4.21) m;jkwm;jkL1 . ; `/ Cm;j . / G . / and (4.26), um;j x ˙ D˙ 2 sin mx and the gap function is given by m;j x D . / G1 > 2 Cm;j sin mx with m;j D Cm;j. We now prove (4.14) with j .From(4.21), Á Á Z Ä q q 1=2 2 1=2C 2 kw ; k 1 1=2 sin y m;j m 1=2 sinh y m;j m m j L > .1 / 2 ˇ Áˇ .1 / 2 Á m;j C m ˇ q ˇ m;j m q dy 4 0 ˇ ` 1=2 2 ˇ ` 1=2C 2 sin m;j m sinh m;j m q = 1=2 2 <` where D m;j m in view of (4.23). By computing the integrals we get

1=2 1=2 1=2 kw ; k 1 C.1/m2 C.1/m2 C.1/m2 m j L > 2 mq;j mq;j D mq;j : 4 1=2 2 1=2 2 1=2 2 m;j m m;j m m;j m

By using (4.23) several more times, we then infer that

>0 1=2 > `3: 9c s.t. m;j kwm;jkL1 c (4.27)

Let us prove the converse inequality. From (4.21) we infer that

1=2 1=2 1 .1 / 2 .1 / 2 kwm;jkL m;j m m;j C m 2j 6 q C q ˇ q Áˇ: 4 1=2 2 1=2 2 ˇ 1=2 ˇ ; Cm ; m ˇ ` 2 ˇ m j m j sin m;j m ˇ q Áˇ ˇ 1=2 ˇ We claim that there exists >0W ˇ sin ` m2 ˇ >for all `>0. If not, q m;j !.`/ ` 1=2 2 . 1/ 0 ` 0 up to a subsequence WD m;j m j ! as ! . By replacing into (4.22), we obtain !.`/ tanh. .j 1//, a contradiction. With this claim,

1=2 1=2 kw ; k 1 .1/m2 C.1/m2 m j L 6 mq;j 2j mq;j : 1=2 C 1=2 4 2 2 m;j Cm m;j m

By using (4.23) several more times, we then infer that

>0 1=2 6 `3: 9c s.t. m;j kwm;jkL1 c (4.28)

Finally, by combining (4.26) with (4.27) and (4.28), we obtain (4.14)forj > 2.The case j D 1 is simpler. In both cases (4.20) and (4.24), wm;1 does not change sign. 4 Torsional Performances of Rectangular Plates 45

Therefore, we do not need to restrict to .0; /. The estimates are then similar to the case j > 2 for (4.27). For (4.28) we proceed as for j > 2:if(4.24) holds, the proof is straightforward since only hyperbolic functions are involved while if (4.20) holds,

1=2 kw ;1k 1 .1/m2 1=2 3=4 m L mq;1 2 6 C `Œ ;1 C .1 /m 6 c ;1 4 1=2 2 m m m;1 Cm and we obtain (4.28). This concludes the proof of Theorem 3. P ˛ . ; / . ; / m wm;j.m/ x y Proof of Theorem 4 The function u x y D m2M k k solves (4.2) with f m;j.m/ wm;j.m/ L1 as given in the statement. By the explicit form of the wm;j.m/,see(4.21) and (4.25), we have X X ˛ . ; `/ m Cm;j.m/ . / G. / ˛ . /: u x ˙ D˙ 2 sin mx and x D mCm;j.m/ sin mx m2M m2M

In particular, we get X 1 G 6 j˛mj Cm;j.m/ : m2M

Let N be the number of integers in the set M. By induction one can prove that if ı . ;:::; / f mgm2M is a nonincreasing setP of positive numbers and SN WD fx D x1 xN 2 N >0 6 1 W xm for x 2 M and m2M xm g, then X ı ı : max mxm D mo x2SN m2M

In our case, xm Dj˛mj and ım D Cm;j.m/. Then the proof of Theorem 4 is completed by noting that the function fmo;j.mo/ satisﬁes the assumption of Theorem 4 with ˛ 1 1 mo D and, by Theorem 3,theL -norm of the corresponding gap function is

Cmo;j.mo/.

Acknowledgements The ﬁrst and second author are partially supported by the Research Project FIR (Futuro in Ricerca) 2013 Geometrical and qualitative aspects of PDEs. The third author is partially supported by the PRIN project Equazioni alle derivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni. The three authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

References

[AmEtAl41] Ammann, O.H., von Kármán, T., Woodruff, G.B.: The Failure of the Tacoma Narrows Bridge. Federal Works Agency, Washington, DC (1941) [BeEtAl] Berchio, E., Buoso, D., Gazzola, F.: On the variation of longitudinal and torsional frequencies in a partially hinged rectangular plate, ESAIM: COCV, doi:10.1051/ cocv/2016076 46 E. Berchio et al.

[BeEtAl16] Berchio, E., Ferrero, A., Gazzola, F.: Structural instability of nonlinear plates modelling suspension bridges: mathematical answers to some long-standing questions. Nonlinear Anal. Real World Appl. 28, 91–125 (2016) [FeEtAl15] Ferrero, A., Gazzola, F.: A partially hinged rectangular plate as a model for suspension bridges. Discrete Contin. Dyn. Syst. A 35, 5879–5908 (2015) [Ga15] Gazzola, F.: Mathematical Models for Suspension Bridges. MS&A, vol. 15. Springer, Cham (2015) [Ki50] Kirchhoff, G.R.: Über das gleichgewicht und die bewegung einer elastischen scheibe. J. Reine Angew. Math. 40, 51–88 (1850) [Lo27] Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Cambridge University Press, Cambridge (1927) [Ma05] Mansﬁeld, E.H.: The Bending and Stretching of Plates, 2nd edn. Cambridge University Press, Cambridge (2005) [Sc01] Scott, R.: In the Wake of Tacoma. Suspension Bridges and the Quest for Aerody- namic Stability. ASCE Press, Reston (2001) Chapter 5 On a Class of Integral Equations Involving Kernels of Cosine and Sine Type

L.P. Castro, R.C. Guerra, and N.M. Tuan

5.1 Introduction

Integral equations involving kernels of cosine and sine type play a significant role in the modelling of different kinds of applied problems. This is the case e.g., when using information-bearing entities like signals which are usually represented by functions of one or more independent variables [OzEtAl01]. The possibilities of application are very diverse, but it is not our intention to present here their description. For this we just refer the interested reader, e.g., to the monograph [OzEtAl01]. Anyway, we would like to observe that this occurs also in the larger class of the so-called linear canonical transforms, where several kernels and integral transforms can be agglutinated into a single integral operator, and so into a consequent integral equation, opening the possibility of defining very flexible classes of integral equations which are quite useful in applications. This paper is divided into three sections. After this introduction, the second section is concerned with the proof of a polynomial identity, for the cosine and sine type integral operator T, defined below by (5.2), and with the solvability of the integral equations of the type (5.1), generated by T.Akeystepinthis section is the construction of certain projection operators, by which our initial integral equation (5.1) may be equivalently transformed into a new integral equation (see (5.8)), governed by special projections. The third section is concentrated on the analysis of properties of the operator T. In particular, we will characterize its spectrum, its invertibility property, derive a formula for its inverse, and obtain

L.P. Castro () • R.C. Guerra CIDMA, University of Aveiro, Aveiro, Portugal e-mail: [email protected]; [email protected] N.M. Tuan Vietnam National University, Hanoi, Vietnam e-mail: [email protected]

© Springer International Publishing AG 2017 47 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_5 48 L.P. Castro et al. a corresponding Parseval-type identity. The 3-order involution property of the operator T is also presented, which is remarkably different from that one of some well-known integral operators such as the Fourier, Cauchy, Hankel and Hilbert integral operators. Moreover, a new convolution and a consequent factorization identity are obtained – allowing, therefore, further studies for associated new convolution type integral equations, as well as eventual new applications; cf. [AnEtAl16, CaSp00, CaZh05, GiEtAl09] and the references cited there.

5.2 Integral Equations Generated by an Integral Operator with Cosine and Sine Kernels

Within the framework of L2.n/, we will consider integral equations of the type

˛' C ˇT' C T2' D g; (5.1) where ˛; ˇ; 2 ¼, with j˛jCjˇjCj j¤0; and g 2 L2.n/ are the given data, and the operator T is deﬁned, in L2.n/,by

2; ; ; ¼; ; 0; T WD aI C bTs C cTc a b c 2 b ¤˙c bc ¤ (5.2) with Z Z 1 1 .T f /.x/ .xy/f .y/dy;.T f /.x/ .xy/f .y/dy: s WD n=2 sin c WD n=2 cos .2 / n .2 / n

We recall the concept of algebraic operators – which will be an important tool in what follows. Let X be a linear space over the complex ﬁeld ¼; and let L.X/ be the set of all linear operators with domain and range in X. Deﬁnition 1 (see [PR88, PR01]) An operator K 2 L.X/ is said to be algebraic if m m1 there exists a normed (non-zero) polynomial P.t/ D t C˛1t CC˛m1tC˛m, ˛j 2 ¼, j D 1;2;:::;m such that P.K/ D 0 on X. We say that an algebraic operator K 2 L.X/ is of order m if there does not exist a normed polynomial Q.t/ of degree k < m such that Q.K/ D 0 on X.In this case, P.t/ is called the characteristic polynomial of K, and the roots of this polynomial are called the characteristic roots of K. In the sequel, the characteristic polynomial of an algebraic operator K will be denoted by PK.t/. Algebraic operators with a characteristic polynomial tm 1 or tm C 1 (m 2) are called involutions or anti-involutions of order m, respectively. An involution (or anti-involution) of order 2 is called, in brief, involution (or anti-involution). In appropriate spaces, most of the important integral operators are algebraic operators. For instance, the Hankel operator J, the Cauchy singular integral operator S on a closed curve, and the 2 Hartley operator H are involutions of order 2, i.e., PJ.t/ D PS.t/ D PH .t/ D t 1 5 On a Class of Integral Equations 49

4 and the Fourier operator F is an involution of order 4, with PF.t/ D t 1.On 2 the other hand, the Hilbert operator H is an anti-involution, as PH.t/ D t C 1 (see [Br86, Ga90, Li00]). Algebraic operators possess some properties that are very useful for solving equations somehow characterized by these operators. Several kinds of integral, ordinary and partial differential equations with transformed arguments can be identiﬁed in such a class of operators (see [PR88, PR01]). Lemma 1 is useful for proving Theorem 1. R 1 . /Œ . /=. / Lemma 1 ([Ti86, Theorem 12]) The formula lim f t sin x t x t !1 dt D .f .x C 0/ C f .x 0//=2 holds if f .x/=.1 Cjxj/ belongs to L1./: We start with a theorem on the characteristic polynomial of the operator T. Theorem 1 The operator T (presented in (5.2)) is an algebraic operator, whose characteristic polynomial is given by

3 2 2 2 2 2 PT .t/ WD t .3a C c/t C .3a b C 2ac/t .a b /.a C c/: (5.3)

Thus, T fulﬁlls the operator polynomial identity:

T3 .3a C c/T2 C .3a2 b2 C 2ac/T .a2 b2/.a C c/I D 0: (5.4)

2 2 3 3 Proof Firstly, we shall prove that Tc C Ts D I, Tc D Tc and Ts D Ts (see [PR88]). Indeed, we have that .Tcf /.x/ D .Tcf /.x/; .Tsf /.x/ D.Tsf /.x/, and TcTs D n TsTc D 0.For>0;let B.0; / WD fy D .y1;:::;yn/ 2 WjykjÄ;8k D 1;:::;ng be the n-dimensional box in n: In the sequel, we will denote by S the Schwartz space. Let f 2 S be given. We can prove inductively on n that Z n 2 sin..x1 t1// sin..x t // cos.y.x t//dy D n n : (5.5) B.0;/ .x1 t1/ .xn tn/

Using (5.5) and Lemma 1,wehave Z Z 1 .T2f /.x/dx D lim f .t/ cos.xy/ cos.yt/ dy dt c n !1 .2 / n B.0;/ Z Z 1 D lim f .t/ Œcos.y.x t// C cos.y.x C t// dy dt n !1 2.2 / n B.0;/ Z Ä n 1 2 sin..x1 t1// sin..x t // f .t/ n n D n lim 2.2 / !1 n .x1 t1/ .xn tn/

n 2 sin..x1 C t1// sin..x C t // f .x/ C f .x/ C n n dt D : .x1 C t1/ .xn C tn/ 2 50 L.P. Castro et al.

. 2 /. / . . / . //=2: Hence, Tc f x D f x C f x By the same way, we are able to prove that . 2 /. / . . / . //=2: . 2 /. / . 2 /. / . /; Ts f x D f x f x Therefore, Tc f x C Ts f x D f x which proves the first identity. . 3 /. / . 2 /. / .. /. / . /. //=2 For the second one, we have Tc f x D Tc Tc f x D Tcf x C Tcf x . /. /: 3 : 3 : D Tcf x So, Tc D Tc Similarly, we have that Ts D Ts The identities are proved for any f 2 S. Note that the space S is dense in L2.n/, and the operators 2 n Tc; Ts can be extended into the Hilbert space L . /. So, we will consider that the operators Tc and Ts are defined in this space and the above identities also hold true 2 n 2 n for all f 2 L . /. Thus, Tc; Ts are algebraic operators in L . / with characteristic . / . / 3 polynomials: PTc t D PTs t D t t (see also [PR88, PR01]). We now define three projections associated with Tc,

2; . 2 /=2 . 2 /=2; Q1 D I Tc Q2 D Tc Tc and Q3 D Tc C Tc (5.6) which satisfy the identities QjQk D ıjkQk,forj; k D 1; 2; 3, Q1 C Q2 C Q3 D I, Tc DQ2 C Q3, and three projections corresponding to Ts,

2; . 2 /=2; . 2 /=2; R1 D I Ts R2 D Ts Ts and R3 D Ts C Ts (5.7) which satisfy RjRk D ıjkRk,forj; k D 1; 2; 3, R1 C R2 C R3 D I, Ts DR2 C R3, where ıjk is denoting the Kronecker delta. Therefore, we are able to rewrite T in terms of orthogonal projection operators:

T D a.Q2 C Q3 C R2 C R3/ C b.R2 C R3/ C c.Q2 C Q3/

D Œ0 Q1 C .a C c/Q2 C .a C c/Q3 C Œ0 R1 C .a b/R2 C .a C b/R3 DW Œ0I a C cI a C cI 0I a bI a C b :

By the above-mentioned identities and some computations, we have that

T3 .3a C c/T2 C .3a2 b2 C 2ac/T .a2 b2/.a C c/I D 0I .a C c/3I .a C c/3I 0I .a b/3I .a C b/3 .3a C c/ 0I .a C c/2I .a C c/2I 0I .a b/2I .a C b/2 C 3a2 b2 C 2ac 0I .a C c/I .a C c/I 0I .a b/I .a C b/ .a2 b2/.a C c/ Œ0I 1I 1I 0I 1I 1 D Œ0I 0I 0I 0I 0I 0 D 0:

It remains to be proven that there does not exist any polynomial Q with deg.Q/<3, and Q.T/ D 0. For that, suppose that there is a such polynomial Q.t/ D t2 C pt C q such that Q.T/ D 0. This is equivalent to 5 On a Class of Integral Equations 51 8 ˆ 2 <ˆ.a C c/ C p.a C c/ C q D 0 . /2 . / 0 ˆ a b C p a b C q D :ˆ .a C b/2 C p.a C b/ C q D 0; whose solutions are b D c D 0 or b D 0 and c ¤ 0 or b D˙c and b ¤ 0.Any solution is not under the conditions b ¤˙c and bc ¤ 0: The polynomial (5.3) has the single roots t1 WD ab, t2 WD aCb and t3 WD aCc. Having this in mind, we are able to build projections, induced by T, in the sense of the Lagrange interpolation formula. Namely:

2 .T t2I/.T t3I/ T .t2 C t3/T C t2t3 P1 WD D I .t1 t2/.t1 t3/ .t1 t2/.t1 t3/ 2 .T t1I/.T t3I/ T .t1 C t3/T C t1t3 P2 WD D I .t2 t1/.t2 t3/ .t2 t1/.t2 t3/ 2 .T t1I/.T t2I/ T .t1 C t2/T C t1t2 P3 WD D : .t3 t1/.t3 t2/ .t3 t1/.t3 t2/

` ` ` ` Then, we have PjPk D ıjkPk; and T D t1P1 C t2P2 C t3P3; for any j; k D 1; 2; 3; and ` D 0; 1; 2. The construction of the just presented projections has the proﬁt of allowing to rewrite (5.1) in the following equivalent way:

' ' ' ; ˛ ˇ 2; 1; 2; 3: m1P1 C m2P2 C m3P3 D g with mj D C tj C tj j D (5.8)

Let 'k.x/ denote the multi-dimensional Hermite functions (see [Th98]). The Hermite functions are essential in several applications, and are also somehow associated with our operator T. Namely, we have (see [TuHu12a, TuHu12b]) 8 ˆ <ˆ'k.x/; if jkjÁ0.mod 4/ . ' /. / 0; 1; 3 . 4/ Tc k x D ˆ if jkjÁ mod (5.9) :ˆ 'k.x/; if jkjÁ2.mod 4/ and 8 ˆ <ˆ0; if jkjÁ0; 2 .mod 4/ . ' /. / ' . /; 1. 4/ Ts k x D ˆ k x if jkjÁ mod (5.10) :ˆ 'k.x/; if jkjÁ3.mod 4/: 52 L.P. Castro et al.

By (5.9)–(5.10), we obtain 8 ˆ <ˆ.a C c/ 'k.x/; if jkjÁ0; 2 .mod 4/ . ' / . / . / ' . /; 1.4/ T k x D ˆ a C b k x if jkjÁ mod (5.11) :ˆ .a b/ 'k.x/; if jkjÁ3.mod 4/:

Therefore, the Hermite functions are eigenfunctions of T with the eigenvalues a ˙ b; a C c. Theorem 2 (i) The integral equation (5.8) (or (5.1)) has a unique solution if and only if m1m2m3 ¤ 0. (ii) If m1m2m3 ¤ 0, then the unique solution of (5.8) is given by

1 1 1 ' D m1 P1g C m2 P2g C m3 P3g: (5.12)

(iii) If mj D 0,forsomejD 1; 2; 3, then the equation (5.8) has solution if and only if Pjg D 0: (iv) If Pjg D 0,forsomejD 1; 2; 3, then the equation (5.8) has an inﬁnite number of solutions given by X X ' 1 ; . /: D mj Pjg C z where z 2 ker Pj (5.13) jÄ3 jÄ3 mj¤0 mj¤0

2 n Proof Suppose that the equation (5.1) has a solution ' 2 L . /. Applying Pj to both sides of the equation (5.8), we obtain a system of three equations mjPj' D Pjg, j D 1; 2; 3: In this way, if m1 m2 m3 ¤ 0, then we have the following equivalent system of equations 8 ˆ 1 <ˆP1' D m1 P1g 1 P2' D m P2g ˆ 2 : 1 P3' D m3 P3g:

Using the identity P1 C P2 C P3 D I; we obtain (5.12). Conversely, we can directly verify, by substitution, that ' given by (5.12) fulfills (5.8). If m1 m2 m3 D 0, then mj D 0,forsomej 2f1; 2; 3g: Therefore, it follows that Pjg D 0: Then, we have X X ' 1 : Pj D mj Pjg jÄ3 jÄ3 mj¤0 mj¤0 5 On a Class of Integral Equations 53 P P P ı . /' . /Œ 1 Using the fact that PjPk D jkPk, we get jÄ3 Pj D jÄ3 Pj jÄ3 mj Pjg 0 0 0 P P mj¤ mj¤ mj¤ . /Œ' 1 0: or, equivalently, jÄ3 Pj jÄ3 mj Pjg D By this, we obtain the mj¤0 mj¤0 solution (5.13). Conversely, we can directly verify that all the elements ', with the form of (5.13), fulfill (5.8). As the Hermite functions are eigenfunctions of T (cf. (5.9)–(5.10)), we conclude that the cardinality of all elements ' in (5.13) is infinite.

5.3 Operator Properties

In this section, we will present some properties of the operator T, deﬁned by (5.2). Namely, we will analyse its invertibility, Parseval-type identity, point spectrum, and the 3-order involution property of T.

5.3.1 Invertibility and Spectrum

In this subsection, we determine the spectrum and characterize the invertibility of T. Moreover, we will explicitly determine the inverse of T (when it exists). Proposition 1 The spectrum of the operator T is given by

.T/ D fa b; a C b; a C cg :

Proof By using (5.11), we deduce that fa b; a C b; a C cg .T/: On the other side, for any 2 ¼,wehave Á Á 3 2 2 2 2 2 t .3a C c/ t C 3a b C 2ac t a b .a C c/ h Ái 2 2 2 2 D .t / t C . 3a c/ t C .3a C c/ C 3a b C 2ac C PT ./; cf. (5.3). Suppose that … fa b; a C b; a C cg : This implies that 3 2 2 2 2 2 PT ./ D .3a C c/ C 3a b C 2ac a b .a C c/ ¤ 0:

Then, the operator T I is invertible, and the inverse operator is deﬁned by 1 1 .T I/ D T2 C . .3a C c// T PT ./ C 2 .3a C c/ C 3a2 b2 C 2ac I :

Thus, we have .T/ D fa b; a C b; a C cg. 54 L.P. Castro et al.

Theorem 3 (Inversion theorem) The operator T, presented in (5.2), is an invertible operator if and only if

a b ¤ 0; a C b ¤ 0 and a C c ¤ 0: (5.14)

In case (5.14) holds, the inverse operator is deﬁned by 1 T1 WD T2 .3a C c/T C .3a2 b2 C 2ac/I : (5.15) .a2 b2/.a C c/

Proof If T is invertible, then it is, at least, injective. Taking into account the Hermite functions 'k, we have already observed that (5.9)–(5.10) hold true. Indeed, by (5.11), we see that the Hermite functions are eigenfunctions of T with eigenvalues a ˙ b and a C c. So, we deduce that .a2 b2/.a C c/ ¤ 0, which is equivalent to (5.14). Conversely, suppose that .a2 b2/.a C c/ ¤ 0. Hence, it is possible to consider the operator deﬁned in (5.15) and, by a straightforward computation, verify that this is, indeed, the inverse of T.

5.3.2 Parseval-Type Identity and Unitary Properties

This subsection is dedicated to obtain conditions which characterize when the operator T is unitary, and to derive a Parseval-type identity associated with T. 2 n Let h; i2 denote the usual inner product of the Hilbert space L . /. Theorem 4 (Parseval-type identity) For any f ; g 2 L2.n/, the identity ˚ « 2 hTf ; Tgi2 Djaj hf ; gi2 C 2Re abN hf ; Tsgi2 2 2 ; 2 2 ; 2 C Re facNg Cjcj hf Tc gi2 Cjbj hf Ts gi2 (5.16) holds. 2 n Proof For any f ; g 2 L . /, a direct computation yields hTsf ; giDhf ; Tsgi, ; ; ; 2 ; 2 0 2 ; ; 2 0 hTcf giDhf Tcgi, hTsf Tc giDhf TsTc giD , hTc f TsgiDhf Tc TsgiD . Thus, (5.16) directly appears simply by using the deﬁnition of T and the just presented identities.

Theorem 5 (Unitary property) Let a WD a1Cia2,bWD b1Cib2,cWD c1Cic2, with a1; a2; b1; b2; c1; c2 2 . T is a unitary operator if and only if one of the following conditions holds: (i) a D 0; jbjD1 and jcjD1; 0 0 2 2 1 . /2 2 1 (ii) a 2 nf g,b1 Dp,a1 C b2 D and a1 C c1 C c2 D ; 2 a2 1jaj a1b1 2 2 2 0 1 2 . 1 1/ . 2 2/ 1 (iii) a ¤ ,b D˙ jaj ,b D a2 and a C c C a C c D . 5 On a Class of Integral Equations 55

Proof Let T be the adjoint operator of T, this is the operator satisfying 2 n hTf ; gi2 Dhf ; T gi2, for any f ; g 2 L . /. From the last identity, we obtain N 2 T DNaI C bTs CNcTc and we know that T is a unitary operator if T is bijective an 1 a N b d T D T . In this way, taking into account (5.15), we have aN D a2b2 , b Da2b2 b2Cac and cN D.a2b2/.aCc/ , whose solutions are (i)–(iii).

5.3.3 Involution

In this subsection, we present conditions under which T is an involution of order 3. This is totally different from the operator structure of other integral operators exhibiting involutions of either 2 or 4 orders. Theorem 6 Consider the operator T deﬁned in (5.2). We have T3 D Ifor p 1 3 5 3 4 (i) a D ei 3 ,bD ei 6 and c D ei 3 ; 2 p2 2 1 3 3 4 (ii) a D ei 3 ,bD ei 6 and c D ei 3 ; 2 2p 2 (iii) a D1 ,bDi 3 and c D 3 ; 2 p 2 2 (iv) a D1 ,bD i 3 and c D 3 ; 2 2 p 2 1 3 3 2 (v) a D ei 3 ,bD ei 6 and c D ei 3 ; 2 p2 2 1 3 5 3 2 i 3 i 6 i 3 (vi) a D 2 e ,bD 2 e and c D 2 e . This means that the operator T is an involution of order 3. Proof If we consider the operator T and its characteristic polynomial (5.3), we have (5.4), which is equivalent to T3.3aCc/T2C.3a2b2C2ac/T D .a2b2/.aC c/I. If we consider .3a C c/ D 0, 3a2 b2 C 2ac D 0 and .a2 b2/.a C c/ D 1, we obtain the solutions (i)–(vi), which means that, for these cases of the parameters, we have T3 D I. Remark 1 Note that the last property helps us to realize that if the coefﬁcients a; b; c are those of the Theorem 6, then our main integral equation, presented before in (5.1), is the most general integral equation that can be generated by the operator T and its powers.

5.3.4 New Convolution

In this subsection we will focus on obtaining a new convolution ~ for the operator T in the case of a D 0. This means that we are introducing a new multiplication operation that has a corresponding multiplicative factorization property for the operator T, in the sense as it occurs for the usual convolution and the Fourier transform: T.f ~ g/ D .Tf /.Tg/. 56 L.P. Castro et al.

Deﬁnition 2 Considering b; c 2 ¼ with b ¤˙c and bc ¤ 0, we deﬁne the new multiplication (convolution) for any two elements f ; g 2 L2.n/: 1 ˚ f ~ gWD b4 T2 Œ.T f /.T g/ .T f /.T2g/ .T2f /.T g/ b2c s s s s c c s 3 2 . /. 2 / 2 . 2 /. / Œ. /. / Cb c Ts Tsf Tc g C Ts Tc f Tsg Ts Tsf Tsg « 2 2 2 . 2 /. 2 / . 2 /. 2 / 3 . 2 /. 2 / : C b c Ts Tc f Tc g Tc f Tc g bc Ts Tc f Tc g (5.17) 2 ; ¼ 0 Theorem 7 For the operator T0 WD bTs C cTc , with b c 2 ,b¤˙c and bc ¤ , 2 n and f ; g 2 L . /, we have the following T0-factorization:

T0 .f ~ g/ D .T0f /.T0g/: (5.18)

Proof Using the deﬁnition of T0 and a direct (but long) computation, we obtain the equivalence between (5.17) and 2 2 2 f ~ g D T0 cT0 b I Œ.T0f /.T0g/=.b c/:

Thus, having in mind (5.15), for a D 0, which provides the formula for the inverse 1 of T0, we identify the last identity with f ~g D T0 Œ.T0f /.T0g/, which is equivalent to (5.18), as desired. We would like to point out that we can also obtain an even more general convolution (in comparison with (5.17)–(5.18)), for T with a ¤ 0, but due to the lack of space we are not presenting it here. We notice that these convolutions, and consequent factorization identities, allow the consideration of new convolution type equations.

Acknowledgements This work was supported in part by Portuguese funds through the CIDMA – Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e a Tecnologia”), within project UID/MAT/04106/2013. The second named author was also supported by FCT through the Ph.D. scholarship PD/BD/114187/2016. The third named author was supported partially by the Viet Nam National Foundation for Science and Technology Developments (NAFOSTED).

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[GiEtAl09] Giang, B.T., Mau, N.V., Tuan, N.M.: Operational properties of two integral transforms of Fourier type and their convolutions. Integr. Equ. Oper. Theory 65, 363–386 (2009) [Li00] Litvinchuk, G.S.: Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Kluwer Academic Publishers, Dordrecht (2000) [OzEtAl01] Ozaktas, H.M., Zalevsky, Z., Kutay, M.A.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2001) [PR88] Przeworska-Rolewicz, D.: Algebraic Analysis. PWN-Polish Scientiﬁc Publishers, Warsawa (1988) [PR01] Przeworska-Rolewicz, D.: Some open questions in algebraic analysis. In: Abe, J.M., Tanaka, S. (eds.) Unsolved Problems on Mathematics for the 21st Century, pp. 109–126, 1st edn. IOS Press, Amsterdam (2001) [Th98] Thangavelu, S.: Harmonic Analysis on the Heisenberg Group. Springer, New York (1998) [Ti86] Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Chelsea, New York (1986) [TuHu12a] Tuan, N.M., Huyen, N.T.T.: The Hermite functions related to inﬁnite series of generalized convolutions and applications. Complex Anal. Oper. Theory 6, 219–236 (2012) [TuHu12b] Tuan, N.M., Huyen, N.T.T.: Applications of generalized convolutions associated with the Fourier and Hartley transforms. J. Integr. Equ. Appl. 24, 111–130 (2012) Chapter 6 The Simple-Layer Potential Approach to the Dirichlet Problem: An Extension to Higher Dimensions of Muskhelishvili Method and Applications

A. Cialdea

6.1 Introduction

The Dirichlet problem for harmonic functions in a bounded domain ˝ n with a sufﬁciently smooth boundary ˙ can be solved by using potential theory in two different approaches. The most classical, which goes back to Fredholm, uses a representation of the unknown function u by means of a double-layer potential Z @ u.x/ D '.y/ s.x; y/ dy I (6.1) ˙ @ y the second one uses a representation of u by means of a simple-layer potential Z

u.x/ D '.y/ s.x; y/ dy : (6.2) ˙

Here we denote by s.x; y/ the fundamental solution of Laplace equation ( 1 log jx yj if n D 2; s.x; y/ D 2 1 jx yj2n if n 3; .n2/!n

n !n being the hypersurface measure of the unit sphere in . If we use (6.1) and impose the boundary condition u D g on ˙, we get a well- known Fredholm equation (see, e.g., [Fo95, Ch.3]).

A. Cialdea () University of Basilicata, Potenza, Italy e-mail: [email protected]

Instead, if we use (6.2), we get an integral equation of the ﬁrst kind Z

'.y/s.x; y/ dy D g.x/; x 2 ˙: (6.3) ˙

In the case n D 2 the first method for solving this equation was given by Muskhelishvili, which reduced (6.3) to a singular integral equation on ˙.This method will be outlined in Section 6.2.1. Muskhelishvili’s method, hinging on the theory of holomorphic functions of one complex variable, cannot be immediately generalized to n >2. The aim of this paper is to present an approach given in [Ci88] which can be viewed as an extension to n of Muskhelishvili’s method. We have to say that nowadays there are different approaches available to discuss equation (6.3). Differently from other methods, ours uses neither the theory of pseudodifferential operators nor the concept of hypersingular integrals. Moreover the use of differential forms immediately leads to interesting results. As an example, see [CiEtAl12, Sec.6], where necessary and sufficient conditions for the existence of a 2-form conjugate to a harmonic function in a multiply connected domain of n are provided and an explicit formula for such differential forms is given. Our approach has been recently used in several boundary value problems for other differential operators (see [CiHs95, CiEtAl11, CiEtAl12, CiEtAl13, CiEtAl13a, CiEtAl14, CiEtAl15, CiEtAl15a, Ma04, Ma05]). Another problem which can be considered using the same method concerns the extension of the classical double-layer potential approach to higher-order operators. A first answer was given by Agmon [Ag57], which considered a higher-order elliptic operator in the plane with constant coefficients and no lower order terms. As Agmon remarks in his paper, his method cannot be used for higher-order equations in more than two variables. Recently, the monograph [MiMi13] appeared. There Irina Mitrea and Marius Mitrea systematically develop a multilayer theory applicable to higher- order partial differential operators with constant coefficients in Lipschitz domains. Both Agmon’s and Mitrea’s method are exhaustive and analytically rich, but they do not seem to give to the solution a simple integral representation like (6.1). We believe that the important monograph [MiMi13] will give a new impulse to this classical question in potential theory and maybe it is not useless to recall a different approach leading to a much simpler integral representation. In the paper [Ci92] a multiple-layer potential theory alternative to Agmon’s was proposed. It concerns the case n D 2 and the same kind of partial differential operators considered by Agmon. This method provides a simple integral representation which can be seen as a generalization of (6.1) and it leads to a singular integral system of regular type. The same kind of integral representation was later used in the biharmonic problem in any number of variables [Ci92a]. In this case the Dirichlet condition leads to a singular integral system, whose symbol determinant identically vanishes. This precludes the possibility of applying the usual theory of pseudodifferential operators. However, in [Ci92a] an existence theorem was given. It hinges on our method for solving (6.3) we discussed below. These results are 6 The Simple-Layer Potential Approach 61 outlined in Section 6.3. Although the equation considered is very particular, it is very likely that the theory may be extended to more general cases. This seems to be an appealing area for future research.

6.2 Muskhelishvili’s Method and Its Extension to n

6.2.1 Muskhelishvili’s Method

In his famous book on singular integral equations [Mu72], Muskhelishvili gave a method for discussing the integral equation of the ﬁrst kind (6.3) in the case n D 2: Z 1 '.y/ log jx yj dsy D g.x/; x 2 ˙: (6.4) 2 ˙

In particular, he was looking for an Hölder continuous solution ' with the data g belonging to C1C˛.˙/ (0<˛Ä 1). The idea of Muskhelishvili is to derive both sides of (6.4) with respect to the arc length s on ˙. In this way we arrive to the singular integral equation: Z 1 @ @g '.y/ log jx yj dsy D .x/; x 2 ˙: (6.5) 2 ˙ @sx @s

By using the theory of holomorphic function of one complex variable, Muskhe- lishvili showed that (6.5) admits an Hölder continuous solution and by this he deduces an existence theorem for equation (6.4). His method cannot be immediately generalized to higher dimensions for two reasons. The ﬁrst one is that we have to understand how to replace the derivative with respect to the arc length in higher dimension. The second one is that - in higher dimension - we cannot apply the theory of holomorphic functions of one complex variable. We have overcome these difﬁculties by using the theory of conjugate differential forms. In the next subsection we outline the ideas of our method and show some applications.

6.2.2 Conjugate Differential Forms

A differential form of degree k (brieﬂy a k-form) deﬁned in an open set ˝ n is given by 1 u D u ::: dxs1 :::dxsk kŠ s1 sk 62 A. Cialdea

::: where us1 sk are the components of a skew-symmetric covariant tensor deﬁned in ˝.Ifu is a k-form and v is an h-form, by u ^ v we denote the .k C h/-form 1 u ::: v ::: dxs1 :::dxsk dxi1 :::dxih : kŠhŠ s1 sk i1 ih

h.˝/ ˝ By Ck we denote the space of k-forms defined in , whose coefficients h.˝/ 1 0 belong to C . The differential d W Ck ! CkC1 is the operator defined by 1 @ ::: j s1 ::: sk : du D us1 sk dx dx dx kŠ @xj

ı 1 0 ı . 1/n.kC1/C1 The co-differential W Ck ! Ck1 is the operator given by u D h h d u, where WCk ! Cnk is the -Hodge (adjoint) operator: 1 u D u ::: dxi1 :::dxink ; .n k/Š i1 ink the coefﬁcients being 1 1:::::::::::n u ::: D ı :: :: u 1::: : i1 ink kŠ s1 ski1 ink s sk

The operators d and ı are related to the Laplacian through the following formula: 1 .dı C ıd/u D u D u ::: dxs1 :::dxsk kŠ s1 sk 0 2 1.˝/ v 1 .˝/ Let Ä k Ä n ; we say that u 2 Ck and 2 CkC2 are conjugate if ( du D ıv (6.6) ıu D 0; dv D 0 in ˝. The system (6.6) provides a generalization of the Cauchy-Riemann system. In fact, if n D 2 we have that u0 and u2 are conjugate forms if, and only if, the scalar function u0 C i. u2/ is holomorphic. The system (6.6) contains, as very particular cases, other system of partial differential equations of the ﬁrst kind which generalize the Cauchy-Riemann system and have been widely studied. For example, if n D 3, 2 3 3 1 1 2 we have that u0 Á u and u2 D v1dx dx C v2dx dx C v3dx dx are conjugate if, and only if, the scalar functions u;v1;v2;v3 satisfy the system: div.v1;v2;v3/ D 0; gradu D curl.v1;v2;v3/ which is known as the Moisil-Theodorescu system (see [MoTh31]). h Consider, in any number of variables, a 1-form u1 D whdx : The form u1 is conjugate to 0 if, and only if, div.w1;:::;wn/ D 0; curl.w1;:::;wn/ D 0; The vectors .w1;:::;wn/ satisfying such a system are called harmonic vectors (see, e.g., [StWe60]). 6 The Simple-Layer Potential Approach 63

More generally, we have that a k-form 1 u D u ::: dxs1 :::dxsk k kŠ s1 sk is conjugate to 0 if, and only if, duk D 0; ıuk D 0: The k-forms which are closed and co-closed are often called harmonic forms.

6.2.3 The Extension to Higher Dimensions of Muskhelishvili Method

Let us consider the integral equation (6.3). Here ˝ is a bounded domain of n with a Lyapunov boundary ˙ (i.e., ˙ 2 C1;, 0<Ä 1). The ﬁrst idea is to replace the derivative with respect to the arc length by the differential (in the sense of the theory of differential forms). In such a way we are led to the integral equation: Z

'.y/dxŒs.x; y/ dy D dg.x/; x 2 ˙: (6.7) ˙

This is not a usual singular integral equation, inasmuch as the unknown is a scalar function while the data is a differential form of degree one. To obtain an existence theorem for such equation, we resort to the theory of reducible operators. We recall that a linear and continuous operator S W B ! B0, where B and B0 are two Banach spaces, can be reduced on the left (on the right) if there exists a linear and continuous operator S0 W B0 ! B such that the operator S0S W B ! B (SS0 W B0 ! B0) is a Fredholm operator, i.e. S0S D ICT (SS0 D ICT), T being compact on B (on B0). The following theorem is well known: Theorem 1 The operator S W B ! B0 can be reduced on the left (on the right) if, and only if, the range S.B/ is closed in B0 and the dimension of the kernel N.S/ (of the kernel of the adjoint N.S/)isﬁnite. If S can be reduced on the left or on the right, its range is closed and we have that there exists a solution of the equation S' D if, and only if, the data is such that

h˚; iD0; 8 ˚ 2 N.S/: (6.8)

We note that, if the operator S can be reduced on the left, the space N.S/ may have infinite dimensions and therefore the number of compatibility conditions (6.8) maybeinfinite. Let now S be the operator given by the left-hand side of (6.7). It can be viewed as p p p a linear and continuous operator from L .˙/ into L1.˙/.HereL1.˙/ denotes the space of differential forms of degree 1 defined on ˙ such that their coefficients are Lp functions. 64 A. Cialdea

In [Ci88] a reducing operator for S is constructed. Speciﬁcally, let sk.x; y/ be the double form introduced by Hodge: X j1 jk j1 jk sk.x; y/ D s.x; y/dx :::dx dy :::dy

j1<:::

In [Ci88]itisprovedthatS0S D ICT, T being a compact operator on Lp.˙/.Itis worthwhile to remark that the proof of this results hinges on the theory of conjugate differential forms. In particular the following formula, holding for any 2 W1;p.˙/ and a.e. on ˙, plays a key role: ÂZ Ã Z @ @ .y/ s.x; y/ dy dx D dx d .y/ ^ sn2.x; y/: @ x ˙ @ y ˙

This is a consequence of the fact that the double layer potential Z @ .y/ s.x; y/ dy ˙ @ y and the 2-form Z

x d .y/ ^ sn2.x; y/ ˙ are conjugate in the sense of Section 6.2.2. Since S can be regularized on the left, its range is closed and then, given 2 W1;p.˙/, there exists a solution ' 2 Lp.˙/ of the equation Z

'.y/dxŒs.x; y/ dy D .x/; a:e: x 2 ˙: ˙ if, and only if, the data satisﬁes the inﬁnite compatibility conditions: Z ^ D 0; 8 2 N.S/: (6.9) ˙ 6 The Simple-Layer Potential Approach 65

q .˙/ On the other hand, one can prove that the form 2 Ln2 satisﬁes the homogeneous singular integral equation Z S Á .y/ ^ dyŒs.x; y/ D 0 ˙ a.e. on ˙ if, and only if, it is weakly closed. Therefore we have p p Theorem 2 Given 2 L1.˙/, there exists ' 2 L .˙/ such that Z

'.y/ dxŒs.x; y/ dy D .x/; a:e:x 2 ˙ ˙ Z 0 q .˙/ if, and only if, ^ D for any 2 Ln2 weakly closed. ˙ Since the right-hand side of equation (6.7) is an exact form, and exact forms are weakly closed, the compatibility conditions (6.9) are automatically satisﬁed. We have thus the existence theorem Theorem 3 Given g 2 W1;p.˙/, there exists ' 2 Lp.˙/ such that Z

'.y/ dxŒs.x; y/ dy D dg.x/; a:e:x 2 ˙: ˙

From this result a simple-layer representation follows for the Dirichlet problem.

6.3 The Multiple-Layer Approach

In [Ag57] Agmon considered the Dirichlet problem for a partial differential operator of higher-order in two independent variables with constant coefﬁcients and no lower order terms 8 ˆX2m @2m ˆ u < ak D 0 in ˝I @xk@x2mk kD0 1 2 (6.10) ˆ @m1 :ˆ u ˙. 0; 1; : : : ; 1/: k m1k D gk on k D m @x1@x2

The multiple-layer potential according to Agmon is Z

u.z/ D 1Œ'./; s.z;/d C 2Œ'./; s.z;/d C˙ 66 A. Cialdea where 1 and 2 are certain bilinear forms, related to the differential operator. As a fundamental solution Agmon takes

s.z;/D Z 1 dw Œ. / . /2n2 Œ. / . / ; 2 Re x w C y log x w C y 2 .2n 2/Š C L.w/ P (6.11) . / 2m 2mk where L w D kD0 akw is the characteristic polynomial and is a rectifiable Jordan curve in the complex half-plane Imz <0that contains all the zeros of L.w/ with negative imaginary part in its interior. Agmon’s approach was generalized to the multidimensional case in the book [MiMi13], where a definition of multiple-layer potential for higher order operators - too much complicated to be written here - is given (see [MiMi13, (1.11)–(1.13)]). In [Ci92] an alternative definition of multiple-layer potential was proposed. It concerns the same kind of partial differential operator considered by Agmon, i.e. the differential operator in (6.10). The integral Z Xm1 @ @m1 . / ' ./ . ;/ u z D k 1 s z ds (6.12) ˙ @ @m k@k kD0 is the multiple-layer potential according to [Ci92]. Here s.z;/ is the same fundamental solution (6.11) considered by Agmon and .z/ is an Hölder continuous unit vector defined on ˙ such that 0. It is obvious why (6.12) is a generalization of the classical double-layer potential (6.1). It must be said that this definition was inspired by the definition of simple-layer potential given by Fichera [Fi61]inthe planar case and later extended to the biharmonic operator in any number of variables in [Ci91]. If we try to solve Dirichlet problem (6.10) by means of the potential (6.12), we get the singular integral system on ˙

Xm1 Z Z bhk.z/ 'k.z/ ahk.z/'k.z/ C d C 'k.z/Mhk.z;/d i ˙ z ˙ kD0 C C n1 D .1/ 2 gh.z/; h D 0;:::;m 1; z 2 ˙ (6.13) where the coefﬁcients ahk and bhk are given by Z 2n2hk w .1.z/w C 2.z// ahk.z/ D Im dw ; C .xwP CPy/ L.w/ Z 2n2hk w .1.z/w C 2.z// bhk.z/ Di Re dw C .xwP CPy/ L.w/ 6 The Simple-Layer Potential Approach 67

(the dots denoting the derivative with respect to the arc length s) and the kernels Mhk are weakly singular. In [Ci92] it is proved that such a system is of regular type, i.e. that detf.ahk bhk/g detf.ahk C bhk/g¤0 on ˙. In other words the symbol determinant of the system (6.13) does not vanish and then the classical theory of singular systems can be applied. This approach was extended to the biharmonic problem in any number of variables in [Ci92a], where the Dirichlet problem is considered 8

By analogy with (6.12) the integral Z Xn @ @ u.x/ D 'k.y/ s.x; y/ dy (6.15) ˙ @ @y kD1 y k is called a multiple-layer potential. Here s.x; y/ is the classical fundamental solution for the biharmonic operator ( Œ! .n 2/.n 4/1jx yj4n if n D 3; 5; 6; : : : s.x; y/ D n 1 .2!4/ log jx yj if n D 4:

When we impose the Dirichlet condition to the multiple-layer potential (6.15)we get the singular integral system

h.x/ k.x/'k.x/C Z @3 C 'k.y/ j.y/ s.x; y/ dy D gh.x/; x 2 ˙ ˙ @xh@yk@yj .h D 1;:::;n/ (6.16)

Of course, if n D 2, this system reduces to (6.13) with D . However, if n 3, the symbol determinant of (6.16) identically vanishes. This makes impossible to apply the usual theory. Moreover, if we denote by S the singular integral operator on the left-hand side of (6.16), we have that both N.S/ and N.S/ have inﬁnite dimensions. Therefore S can be reduced neither on the left nor on the right (see Theo- rem 1). In order to overcome this difﬁculty, in [Ci92a] the concept of bilateral reduction was introduced. Let S be a linear and continuous operator S W B ! B0, B and B0 being two Banach spaces. We say that a bilateral reduction of S is given, if B00 is a 00 0 00 Banach space, S1 W B ! B, S2 W B ! B are linear and continuous operators and

S2SS1 D I C T (6.17) 68 A. Cialdea where T is a compact operator on B00. Even if it is always possible to give trivial bilateral reductions for an operator S ¤ 0, the following theorem makes this concept useful. Theorem 4 If S is a linear and continuous operator and the bilateral reduc- 0 tion (6.17) is given, then the range S.B/ is closed in B if and only if SŒN.S2S/ is closed in B0. In [Ci92a] a suitable bilateral reduction of the singular integral operator S is constructed in such a way the kernel N.S2S/ has ﬁnite dimension. We remark that such a construction heavily hinges on the results obtained for Laplace equation. This implies that the range of S is closed in ŒLp.˙/n and then a solution of (6.16) exists if and only if the data .g1;:::;gn/ is orthogonal to the eigensolutions of the homogenous adjoint system. From this existence result for the system (6.16) a representation theorem for the solutions of (6.14) follows. We refer to [Ci92a]for all the details.

References

[Ag57] Agmon, S.: Multiple layer potential and the Dirichlet problem for higher order elliptic equations in the plane. Commun. Pure Appl. Math. X, 179–239 (1957) [Ci88] Cialdea, A.: Sul problema della derivata obliqua per le funzioni armoniche e questioni connesse. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 12, 181–200 (1988) [Ci91] Cialdea, A.: The simple layer potential for the biharmonic equation in n variables. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. (9) 2, 115–127 (1991) [Ci92] Cialdea, A.: A multiple layer potential theory alternative to Agmon’s. Arch. Ration. Mech. Anal. 120, 345–362 (1992) [Ci92a] Cialdea, A.: The multiple layer potential for the biharmonic equation in n variables. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. (9) 3, 241–259 (1992) [CiHs95] Cialdea, A., Hsiao, G.: Regularization for some boundary integral equations of the ﬁrst kind in mechanics. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19, 25–42 (1995) [CiEtAl11] Cialdea, A., Leonessa, V., Malaspina, A.: Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains. Bound. Value Probl. 2011, 53, 1–25 (2011) [CiEtAl12] Cialdea, A., Leonessa, V., Malaspina, A.: On the Dirichlet and the Neumann problems for Laplace equation in multiply connected domains. Complex Var. Elliptic Equ. 57, 1035–1054 (2012) [CiEtAl13] Cialdea, A., Leonessa, V., Malaspina, A.: On the Dirichlet problem for the Stokes system in multiply connected domains. Abstr. Appl. Anal. 2013, Article ID 765020, 12 pp. (2013). doi:10.1155/2013/765020 [CiEtAl13a] Cialdea, A., Dolce, E., Malaspina, A., Nanni, V.: On an integral equation of the ﬁrst kind arising in the theory of Cosserat. Int. J. Math. 24(5), 1350037, 21 pp. (2013). doi:10.1142/S0129167X13500377 [CiEtAl14] Cialdea, A., Dolce, E., Leonessa, V., Malaspina, A.: New integral representations in the linear theory of viscoelastic materials with voids. Publ. Inst. Math. (Beograd) 96(110), 49–65 (2014) [CiEtAl15] Cialdea, A., Dolce, E., Malaspina, A.: A complement to potential theory in the Cosserat elasticity. Math. Methods Appl. Sci. 38, 537–547 (2015). doi:10.1002/mma.3086 6 The Simple-Layer Potential Approach 69

[CiEtAl15a] Cialdea, A., Leonessa, V., Malaspina, A.: The Dirichlet problem for second order divergence form elliptic operators with variable coefﬁcients: the simple layer potential ansatz. Abstr. Appl. Anal. 2015, Article ID 276810, 11 pp. (2015). doi:10.1155/2015/276810 [Fi61] Fichera, G.: Linear elliptic equations of higher order in two independent variables and singular integral equations, with applications to anisotropic inhomogeneous elasticity. In: Langer, R.E. (ed.) Partial Differential Equations and Continuum Mechanics, pp. 55–80. University of Wisconsin Press, Madison (1961) [Fo95] Folland, G.B.: An Introduction to Partial Differential Equations. Princeton University Press, Princeton (1995) [Ma04] Malaspina, A.: Regularization for integral equations of the ﬁrst kind in the theory of thermoelastic pseudo-oscillations. Appl. Math. Inform. Mech. 9, 2, 29–51 (2004) [Ma05] Malaspina, A.: On the traction problem in Mechanics. Arch. Mech. 57, 6, 479–491 (2005) [MiMi13] Mitrea, I., Mitrea, M.: Multi-Layer Potentials and Boundary Problems for Higher- Order Elliptic Systems in Lipschitz Domains. Lecture Notes in Mathematics, vol. 2063. Springer, Heidelberg (2013) [MoTh31] Moisil, Gr.C., Theodorescu, N.: Fonctions holomorphes dans l’espace. Mathematica V, 142–159 (1931) [Mu72] Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff, Groningen (1972). Reprinted [StWe60] Stein, E.M., Weiss, G.: On the theory of harmonic functions of several variables. Acta Math. 103, 25–62 (1960) Chapter 7 Bending of Elastic Plates: Generalized Fourier Series Method

C. Constanda and D. Doty

7.1 Introduction

Let S be a ﬁnite domain in 2 bounded by a simple, closed, C2-curve @S,letx and y be generic points in S or on @S, and let h0 D const >0, h0 diam S. We assume that the three-dimensional region .S [ @S/ Œh0=2; h0=2 is occupied by a homogeneous and isotropic elastic material with Lamé constants and . In what follows, jx yj is the Euclidean distance between x and y, C0;˛.@S/, ˛ 2 .0; 1/, is the space of Hölder continuous functions on @S, C1;˛.@S/, ˛ 2 .0; 1/,is the space of Hölder continuously differentiable functions on @S, and I is the identity operator. The system of partial differential equations that describes the equilibrium state in the process of bending of an elastic plate with transverse shear deformation with negligible body forces can be written as [Co14]

A.@1;@2/u.x/ D 0; (7.1) where 0 1 2 2 2 2 h C h . C /@1 h . C /@1@2 @1 @ 2 2 2 2 A A.@1;@2/ D h . C /@1@2 h C h . C /@2 @2 ; @1 @2 p T u D .u1; u2; u3/ is a vector characterizing the displacement, and h D h0= 12.

C. Constanda () • D. Doty Department of Mathematics, The University of Tulsa, Tulsa, OK, USA e-mail: [email protected]; [email protected]

Some additional mathematical objects related to our analysis need to be deﬁned at this stage (see [Co14]). Thus, the columns f .i/ of the matrix 0 1 100 f D @ 010A x1 x2 1 form a basis for the space of rigid displacements, D.x; y/ is a matrix of fundamental .j/ 1; 2; 3; solutions for the matrix operator A, D , j D are the columns of D, T .i/ P.x; y/ D T.@y/D.y; x/ is an associated matrix of singular solutions, and P are the columns of P. Here, T.@x/ D T.@1;@2/ is the boundary moment-force operator given by 0 1 2 2 2 2 h . C 2/ 1@1 C h 2@2 h 2@1 C h 1@2 0 @ 2 2 2 2 A T.@1;@2/ D h 2@1 C h 1@2 h 1@1 C h . C 2/ 2@2 0 1 2 ˛@˛

T and D . 1; 2/ is the unit vector of the outward normal to @S.

7.2 The Boundary Value Problem

The Dirichlet problem consists in solving system (7.1)inS with the boundary condition

u.x/ D P.x/; x 2 @S:

This problem has a unique solution, which admits the integral representation Z u.x/ D D.x; y/ .y/ ds.y/ H.x/; x 2 S;

@S where is an unknown density and Z H.x/ D P.x; y/P.y/ ds.y/; x 2 2 n @S

@S satisﬁes Z 0 D D.x; y/ .y/ H.x/; x 2 2 nfS [ @Sg:

@S 7 Generalized Fourier Series Method 73

2 Let @S be a simple, closed, C -curve surrounding S [ @S, and consider a set of .k/ points fx , k D 1;2;:::g densely distributed on @S. Additionally, we denote by F Df .ik/, i D 1; 2; 3; k D 0; 1; 2; : : :g the set of vector functions on @S deﬁned by

.i0/.x/ D f .i/.x/; .ik/.x/ D D.i/.x; x.k//; k D 1;2;::::

Theorem [Co14]ThesetF is linearly independent on @S and complete in L2.@S/. We re-order the elements of F as the sequence

.10/;.20/;.30/;.11/;.21/;.31/;.12/;.22/;.32/; ::: and re-index them: F Df .1/;.2/; :::g.

7.3 Numerical Example

Let S be the disk of radius 1 centered at origin and let (after rescaling) h D 0:5 and D D 1. We choose the boundary data vector function 2 T P.x/ D 2.x1 C 1/; 2x1x2;4x1 1 ; x 2 @S; for which our boundary value problem has the exact solution 2 2 3 2 T u.x/ D 3x1 C x2 C 1; 2x1x2;5x1 x1 x1x2 1 ; x 2 S:

We take the auxiliary curve @S to be the circle of radius 2 centered at the origin. This choice has been made because if @S is too far away from @S, then the set F becomes “less linearly independent.” At the same time, placing @S too close to @S increases the sensitivity of the elements of F to the singularities of D and P. Obviously, the accuracy of the approximation depends on the selection of the .k/ dense set fx 2 @S; k D 1;2;:::g: In this context, we feel that a reasonably uniform choice would be

.1/ .2/ .3/ fx ; x ; x ;:::gCartesian Df.2; 0/; .2; /; .2; =2/; .2; 3 =2/; .2; =4/; : : :gPolar:

The orthonormalization of F that produces the set FON is done by three different procedures: the classical Gram–Schmidt (CGS), the modiﬁed Gram– Schmidt (MGS), and the Householder reﬂections (HR). These methods have some common features. For example, for every value n D 1;2;:::; each of them generates a traditional QR decomposition

D ˝R;D . .1/ ::: .n//; ˝ D .!.1/ ::: !.n//; 74 C. Constanda and D. Doty where R D .rik/ is an upper-triangular nonsingular matrix. Consequently, we have 1 1 ˝ D R , where R D .rQik/ is also upper triangular. Expressed individually, this means that

Xk .k/ .i/ ! D rQik ; k D 1;:::;n: iD1

We seek an approximation of the unknown density in the form Â Ã Xn Xn Xk Xk .n/ .k/ .k/ .i/ .j/ D h! ; i! D rQikh ; i rQjk ; n D 1;2;:::: kD1 kD1 iD1 jD1

Then the approximate solution is Z u.n/.x/ D D.x; y/ .n/.y/ ds.y/ H.x/; x 2 S:

Given our boundary data function, we compute Z Z P.i/.x.k/; y/P.y/ ds.y/ D H .i/.x.k// D D.i/.x.k/; y/ .y/ ds.y/ Dh .ik/; i:

@S @S

Hence, after re-indexing, we can write h .k/; iD0; k D 1; 2; 3; h .3kCj/; iDH .j/.x.k//; j D 1; 2; 3; k D 1;2;::::

In this numerical example, ﬂoating-point computation is performed with machine precision of approximately 16 digits. As expected, the most sensitive element in the procedure is the evaluation of integrals in inner products, for which we set a target of 11 signiﬁcant digits. However, as computation proceeds, this number deteriorates.

7.4 Graphical Illustrations

The graphs of the components of the term u.51/ computed from .51/ with MGS for 0 Ä r Ä 0:95, 0 Ä ˛<2 , together with the graph of the prescribed function P on @S are shown in Figure 7.1. We limited the range to r Ä 0:95 because of the increasing influence on the result of the singularities of D.x; y/ and P.x; y/ for x 2 S very close to y 2 @S.This problem can be mitigated by increasing the floating-point accuracy in the vicinity of @S, but it can never be completely eliminated. The gap between the computed subdomain and @S is to be filled by appropriate interpolation. 7 Generalized Fourier Series Method 75

Fig. 7.1 The components of u.51/ and P

Fig. 7.2 The relative percentage error in u.51/

Fig. 7.3 Components of typical vector functions .k/ and !.k/

Figure 7.2 shows the graphs of the components of the relative percent error in u.51/. The approximation is between 9 to 11 digits of accuracy away from @S.As can be seen, the accuracy deteriorates signiﬁcantly close to the boundary. The spike at .1; 0/ is caused by the non-enforcement of periodicity at ˛ D 0, ˛ D 2 . .k/ .k/ Some typical three-component vector functions 2 F and ! 2 FON are graphed in Figure 7.3. The increasing number of oscillations in the latter eventually produces computational difﬁculties in the evaluation of inner products occurring in the construction of additional orthonormal vector functions.

7.5 The Classical Gram–Schmidt Procedure (CGS)

In this method, each vector function .k/ is projected on to the space orthogonal to spanf!.1/; :::; !.k1/g. The projection operators are Xk1 . .1// .1/; . .k// . .k// !.i/;.k/ !.i/; 2;:::; ; P1 D I P!1 :::!k1 D I h i k D n iD1 76 C. Constanda and D. Doty

Fig. 7.4 Density plot of the difference between the matrix of the inner products and the identity matrix for i D 1;:::;99in CGS

.1/ .1/ .k/ P1 I P! :::! !.1/ D D ;!.k/ D 1 k1 ; k D 2;:::;n; .1/ .1/ .k/ kP1 k kI k kP!1:::!k1 k and the entries of the matrix R are

.1/ ; .k/ ; 2;:::; ; r11 DkI k rkk DkP!1:::!k1 k k D n .i/ .k/ rik Dh! ; i; i D 1;:::;k 1; k D 2;:::;n;

rik D 0; i D k C 1;:::;n; k D 1;:::;n 1:

In matrix notation, the QR decomposition can be written as D ˝R. The obvious drawback of this method is that, as calculation proceeds, .k/ has the tendency to become increasingly “parallel” to its projection on span f!.1/;:::;!.k1/g, so the difference between them becomes increasingly smaller in magnitude, which leads to computational ill-conditioning. The set of all the inner products h!.i/;!.j/i should generate the identity matrix. To visualize the inaccuracy in the orthonormalization process, we draw a density plot of the difference between the actual matrix and the identity matrix. Theoretically, the result should be identically zero. But, as seen in Figure 7.4,the graph for CGS with f!.i/, i D 1;:::;99g shows that the method breaks down after about 50 vector functions have been orthonormalized. In this ﬁgure, white means 0 and black means 1 (that is, complete loss of orthogonality). In correct orthonormalization, the “angle” between !.i/ and !.j/, i ¤ j, should be =2, but this angle diminishes when the process deteriorates. The graph of the absolute minimum “angle” between distinct !.i/ and !.j/ for the n n sub-matrices with 1 Ä n Ä 99 is illustrated in Figure 7.5. The process breaks down suddenly and severely once inaccuracy takes over. 7 Generalized Fourier Series Method 77

Fig. 7.5 Graph of the absolute minimum “angle” between distinct !.i/ and !.j/ in CGS

7.6 The Modiﬁed Gram–Schmidt Procedure (MGS)

Due to the mutual orthogonality of the !.i/, the projection operator of CGS can be re-factored in the form

; ::: ; 2;:::; : P1 D I P!1:::!k1 D P!k1 P!1 k D n

The MGS is now constructed as .1/ .1/ !.1/ P1 I ; D .1/ D .1/ kP1 k I .k/ .k/ P! :::! P! :::P! !.k/ D 1 k1 D k1 1 ; k D 2;:::;n: .k/ ::: .k/ kP!1:::!k1 k kP!k1 P!1 k

MGS also generates a QR decomposition D ˝R, where R D .rik/ is upper triangular and nonsingular. Theoretically, R is the same for both CGS and MGS; that is,

!.i/;.k/ !.i/; ::: .k/ : h iDrik Dh P!k1 P!1 i

However, computationally they are significantly different. The left-hand side above is computed with the full magnitude of the .k/.The right-hand side is computed with a vector function of much smaller magnitude, which prevents the magnitude of the difference of the two in the inner product from becoming too small. This explains the computational superiority of MGS over CGS when floating-point arithmetic is used. Again, the density plot of the difference between the inner product matrix and the identity matrix should theoretically show a zero result. The graph of the actual result for f!.i/, i D 1;:::;99g can be seen in Figure 7.6. The orthogonalization is quite good throughout and shows only a slight sign of breaking down. As in the CGM, the minimum “angle” between distinct vectors !.i/ and !.j/ should be close to =2. This graph for the n n sub-matrices with 1 Ä n Ä 99 is displayed in Figure 7.7. The first signs of deterioration appear at about n D 96. Ill-conditioning increases steadily in the construction of the orthogonal decomposition as computation progresses. The relationship in MGS between the condition 78 C. Constanda and D. Doty

Fig. 7.6 Density plot of the difference between the matrix of the inner products and the identity matrix for i D 1;:::;99in MGS

Fig. 7.7 Graph of the minimum “angle” between distinct !.i/ and !.j/ for 1 Ä n Ä 99 in MGS

Fig. 7.8 Relationship in MGS between the condition number and the size of matrix R for n D 1;:::;99

number and the size of the n n matrix R for n D 1;:::;99 is illustrated in Figure 7.8. The graph suggests that the onset of deterioration in the computation may be due to ill-conditioning rather than the orthogonalization method.

7.7 The Householder Reﬂection Procedure (HR)

A significant computational shortcoming of the CGS and MGS methods is that the orthogonal projections P!1:::!k1 are not invertible. A Householder reflection Hk is 1 invertible and orthogonal in the sense that Hk D Hk; it is also norm preserving: kHkkDkk. This is therefore the method of choice when performing orthogonal decompositions. Unfortunately, it does not extend in a natural way to infinite- dimensional function spaces. Recently (see [Tr09] and [GoEtAl15]), the HR method has been reinterpreted in order to overcome this constraint. 7 Generalized Fourier Series Method 79

As a ﬁrst step in the procedure, a target orthonormal set of vector functions fe.1/; :::; e.n/g is selected, and the HR decomposition is deﬁned as

.1/ .2/ .n/ .1/ .n/ .H1 ; H2H1 ; :::; Hn :::H2H1 / D ER; E D .e ::: e /:

All we need now is the structure of the Hk and the formulas to compute the upper- triangular matrix R D .rik/. We impose the additional restriction, imported from the ﬁnite-dimensional HR, that

.1/ .k1/ Hk D I on spanfe ;:::;e g:

This leads to a QR decomposition ˝R via a Householder decomposition of a type that we call HQR, denoted in our speciﬁc case by HER:

D H1 :::Hn ER D HER D ˝R:

.k/ .k/ Starting with 0 D , we deﬁne consecutively for 1 Ä k Ä n the sequence of functions

.k/ .k/ ; .i/; .k/ ;1 < ; i D Pe.i/ Hi i1 rik Dhe Hi i1i Ä i k Ä n .k/ . / . / hv ;i v.k/ D k k k ke.k/; H ./ D I./ 2 ;1Ä k Ä n; k1 k1 k kv.k/k2 .k/ .k/ .k/ .k/; .k/ ;1 ; D Hk k1 D˙k k1ke rkk D˙k k1k Ä k Ä n .k/ 0; 0; 1 < : i D rik D Ä k i Ä n

The sign C or is chosen to maximize kv.k/k, which improves the computational accuracy. Remark CGS and MGS start with an incoming set f .1/;:::;.n/g and predeter- mined rules for computing R, and produce an orthonormal set f!.1/;:::;!.n/g. The computational accuracy of the orthogonalization process deteriorates because each subsequent !.k/ depends completely on the accuracy of the preceding !.i/, 1 Ä i Ä k. HR starts with both an incoming set f .1/;:::;.n/g and a preselected .1/ .n/ orthonormal set fe ;:::;e g, and computes the rules Hk and R that transform the former into the latter. Since the e.k/ are known exactly, this orthogonalization process never breaks down. Any computational inaccuracy in the HR comes from the eventual ill-conditioning produced by the construction of the matrix R and associated transformations Hk. 80 C. Constanda and D. Doty

The HR method applied to our example produces an HQR decomposition. The approximation of the unknown density is given by Â Ã Xn Xn Xk Xk .n/ .k/ .k/ .i/ .j/ D h! ; i! D rQikh ; i rQjk ; n D 1;2;:::; kD1 kD1 iD1 jD1

1 1 with R D .rik/ D .rQik/ computed directly as shown earlier, and h .k/; iD0; k D 1; 2; 3; h .3kCj/; iDH .j/.x.k//; j D 1; 2; 3; k D 1;2;::::

Our speciﬁc choice of orthonormal vector functions e.k/, which allows for periodicity on 0 Ä ˛<2 ,is p p p 1= 2 ;0;0 T; 0; 1= 2 ; 0 T; 0; 0; 1= 2 T; p p p .sin ˛/= ;0;0 T; 0; .sin ˛/= ;0 T; 0; 0; .sin ˛/= T; p p p .cos ˛/= ;0;0 T; 0; .cos ˛/= ;0 T; 0; 0; .cos ˛/= T; p p p .sin.2˛//= ;0;0 T; 0; .sin.2˛//= /;0 T; 0; 0; .sin.2˛//= T; p p p .cos.2˛//= ;0;0 T; 0; .cos.2˛//= ;0 T; 0; 0; .cos.2˛//= T; : : : : : :

The approximation u.51/ computed from .51/ by means of HR is virtually indistinguishable from that computed with MGS as both use the same deﬁnite integrals and numerical integration procedures. We have used the HQR decomposition to avoid the expensive computation of the orthonormal vector functions !.k/, k D 1;:::;n.Ifthe!.k/ are needed, they can be computed from the QR decomposition. As seen in Figure 7.9, the density plot of the difference between the matrix of all the !.k/ inner products and the identity matrix shows excellent orthogonality for the !.k/, k D 1;:::;99.

Fig. 7.9 Density plot of the difference between the matrix of the inner products and the identity matrix for k D 1;:::;99in HR 7 Generalized Fourier Series Method 81

Fig. 7.10 Graph of the minimum “angle” between distinct !.i/ and !.j/ for 1 Ä n Ä 99 in HR

The minimum “angle” between distinct !.i/ and !.j/ in the n n sub-matrices with 1 Ä n Ä 99, which should be close to =2, is graphed in Figure 7.10 and shows no deterioration for HR.

References

[Co14] Constanda, C.: Mathematical Methods for Elastic Plates. Springer, London (2014) [GoEtAl15] Gorodetsky, A., Karaman, S., Marzouk, Y.: Function-train: a continuous analogue of the tensor-train decomposition, preprint arXiv: 1510.09088 (2015) [Tr09] Trefethen, L.N.: Householder triangularization of a quasimatrix. IMA J. Numer. Anal. 30, 887–897 (2009). doi:10.1093/imanum/drp018 Chapter 8 Existence and Uniqueness Results for a Class of Singular Elliptic Problems in Two-Component Domains

P. Donato and F. Raimondi

8.1 Introduction

In this work we study the existence, the uniqueness, and the regularity for a weak solution u of a quasilinear elliptic singular problem posed in a two-component N domain ˝ D ˝1 [ ˝2 [ of R , N 2, with an imperfect contact on the interface . More precisely, we focus on the following problem described in Section 8.2: 8 ˆ div.B.x; u/ru/ C u D f .u/ in ˝ n ; ˆ ˆ < u D 0 on @˝; (8.1) ˆ . . ; / / . . ; / / ; ˆ B x u1 ru1 1 D B x u2 ru2 1 on :ˆ .B.x; u1/ru1/ 1 Dh.u1 u2/ on ; where 1 is the unit external normal vector to ˝1, and we prescribe a Dirichlet condition on the exterior boundary and a jump of the solution on the interface. The quasilinear term consists in a uniformly elliptic and bounded Carathéodory matrix ﬁeld B.x; t/, is a nonnegative real number, .s/ is a nonnegative real function singular at s D 0, and f is a nonnegative datum whose summability depends on the growth of near the singularity. As for the boundary condition, the term h appearing

P. Donato () University of Rouen Normandie, Rouen, France e-mail: [email protected] F. Raimondi University of Salerno, Salerno, Italy University of Rouen Normandie, Rouen, France e-mail: [email protected]

© Springer International Publishing AG 2017 83 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_8 84 P. Donato and F. Raimondi in the proportionally coefficient between the continuous heat flux and the jump of the solution is assumed to be bounded and nonnegative on . The difficulty arises in dealing simultaneously with the quasilinear matrix field, the singular datum, and the interface boundary condition. Quasilinear diffusion matrix fields describe the behavior of materials like glass or wood, in which the heat diffusion depends on the range of the temperature (see, for instance, [DoEtAl17]). The boundary condition on models a jump of the solution through a rough interface and we refer to [CaEtAl47] for a physical justification of this model (see also [DoEtAl17, Mo03]). A source term that depends on the solution itself and becomes infinite when the solution vanishes can model, for instance, an electrical conductor where each point becomes a source of heat as a current flows in it. We refer to [DoEtAl17, Section 3] for details on the physical meaning of this problem. The main results are stated in Section 8.4. In order to prove the existence of a solution for our problem, we approximate it with a sequence of nonsingular problems with a bounded nonlinearity in the equation. Then, suitable a priori estimates, together with accurate measure arguments, allow us to pass to the limit in the approximate problem, as done in [DoEtAl17, GiEtAl16] and [GiEtAl17], which present a similar singular term. For the uniqueness of the solution, we need to require additional assumptions on the matrix field B and on the function . In particular we prescribe that is a nonincreasing function, observing that (as in [DoEtAl16]) we do not assume any monotonicity property for it except for this result. We also state the boundedness of a solution assuming stronger hypothesis on the summability of the datum f . Let us point out that the main novelty with respect to the case studied in [DoEtAl16] concerns the proofs of the a priori estimates, given here in detail in Section 8.3. Then, we describe the approximate problem and, for the sake of brevity, merely sketch the remaining proofs. They are obtained by adapting to the present case the corresponding ones from [DoEtAl16], with obvious modifications concerning the boundary term.

8.2 Setting of the Problem

To describe the two-component domain, we suppose N 2 and denote by (Fig. 8.1) • ˝ a connected bounded open set in N and @˝ its boundary, • ˝ WD ˝1 [ ˝2 [ the two-component domain, where ˝2 is an open set such that ˝2 ˝ with a Lipschitz continuous boundary and ˝1 WD ˝ n ˝2, • i the unit outward normal to ˝i for i D 1; 2, ˝ ˝ • ui D uj˝i the restriction to i of a function u deﬁned in , • the zero extension to the whole of ˝ of functions deﬁned on ˝1 and ˝2, 8 Singular Elliptic Problems in Two-Component Domains 85

Fig. 8.1 The two-component ¶W ˝ domain G

W2 W1

N • E the characteristic function of any subset E of , 1 NN • M .˛; ˇ; E/ the set of matrix ﬁelds A D .ai;j/1Äi;jÄN 2 .L .E// such that

.A.x/; / ˛jj2 and jA.x/jÄˇjj; 8 2 N and a.e. on E:

In this paper we study problem (8.1) under the following hypotheses: N . ; / ˝ . ; / ; . ; / H1) The real N N matrix ﬁeld B W x t 2 7! B x t D bi j x t i;jD1 2 2 N is a Carathéodory function such that B.; t/ 2 M.˛;ˇ;˝/; for every t 2 I H ) C 0.Œ0; Œ/; 0 .s/ 1 s 0; Œ; 0< 1 2 2 C1 Ä Ä s for every 2 C1 with Ä I 0; 0 0 ˝; l.˝/; 2 . 1/ H3) Ä f ¥ a.e. in with f 2 L for l 1C I 1 H4) h 2 L . / and there exists h0 2 such that 0

1 v .˝ / v 0 @˝ v v 2 : V1 Df 2 H 1 W D on g with k kV1 WD kr kL .˝1/

1 Let us deﬁne V WD fv Á .v1;v2/ W v1 2 V1 and v2 2 H .˝2/g, equipped with the norm

2 2 2 2 v v1 2 v2 2 v1 v2 2 : k kV WD kr kL .˝1/ Ckr kL .˝2/ Ck kL . / (8.2)

e e 2 2 2 v v1 v2; v v v1 v2 : We identify r WD r C r so that k kV Dkr kL2.˝n/ Ck kL2. /

Remark 1 A Poincaré inequality holds true in V1, i.e., there exists a positive constant Cp such that

v 2 v 2 ; v : k 1kL .˝1/ Ä Cpkr 1kL .˝1/ for every 1 2 V1

We recall also the following equivalence of the norms proved in [Mo03]: 1 Proposition 1 The norm given in (8.2) is equivalent to the norm of V1 H .˝2/, i.e. there exist two positive constants c1 and c2 such that

v v 1 v ; v : c1k kV Äk kV1H .˝2/ Ä c2k kV 8 2 V 86 P. Donato and F. Raimondi

Furthermore let us recall the classical decomposition u D uC u for every function u 2 V, where uC and u are the nonnegative functions deﬁned by

uC WD maxfu;0g; u WD minfu;0g a.e. in ˝:

In particular, for every v 2 V and also for their traces on one has

C C 2 .v1 v2/.v1 v2 / D .v1 v2 /.v1 v2 / .v1 v2 /

C C 2 Dv1 v2 v1 v1 .v1 v2 / Ä 0:

In the sequel we will also use the two real functions introduced below. For every k >0, Tk denotes the usual truncation function at level k. Observe that, if ' 2 V, then Tk.'/ converges to ' almost everywhere in ˝ so that, via the Lebesgue dominated convergence theorem,

Tk.'/ ! ' strongly in V; as k !1: (8.3)

Let also Zı be, for every ı>0, the auxiliary function deﬁned by 8 ˆ 1; if 0 Ä s Ä ı; <ˆ . / s 2; ı 2ı; Zı s D ˆ C if Ä s Ä (8.4) :ˆ ı 0; if s 2ı:

Observe that 0 Ä Zı.s/ Ä 1 for any positive real s and Â Ã 1 Zı.s/ ı 2ı s : r D f ÄsÄ g ı r (8.5)

Our aim is to prove an existence result for problem (8.1), whose variational formulation is the following: 8 ˆ Find u 2 V such that u >0 a.e. in ˝; ˆ Z ˆ ˆ ˆ f .u/'dx < C1 and <ˆ ˝ Z Z Z ˆ (8.6) ˆ B.x; u/rur'dx C u'dx C h.u1 u2/.'1 '2/d ˆ ˝ ˝ ˆ n ˆ Z ˆ : D f .u/'dx; 8' D .'1;'2/ 2 V: ˝

A function u satisfying (8.6) is called a solution to problem (8.1). 8 Singular Elliptic Problems in Two-Component Domains 87

8.3 A Priori Estimates

In this section, we prove some a priori estimates for a solution u to problem (8.1). As usual in the literature, these estimates are the main tool when proving the existence of u. Indeed, since they also apply to a solution um of the approximate problem (8.1), where they are uniform with respect to m, they allow to pass to the limit.

Proposition 2 Under assumptions H1)–H4), let u 2 V be a solution to problem (8.1). The following a priori estimates hold true:

1 1C ; krukL2.˝n/ Ä C1kf k 2 (8.7) L 1C .˝/ where C1 depends on ˛ and Cp. Also

1 1C ; ku1 u2kL2. / Ä C2kf k 2 (8.8) L 1C .˝/ where C2 depends on ˛,h0 and Cp. Proof Let u 2 V be a solution of problem (8.1). Choosing u as test function in (8.6), using H1)–H4) and applying the Young inequality we get Z ˛ 2 . /2 krukL2.˝n/ C ho u1 u2 d Z Z Z 2 2 Ä B.x; u/rurudx C u dx C h.u1 u2/ d Z˝n Z ˝ 2 . / 1 2 ./ 1C : D f u udx Ä fu dx Ä kukL2.˝/ C c kf k 2 ˝ ˝ L 1C .˝/

From Remark 1 we obtain Z 2 2 2 2 1C ˛ 2 . 1 2/ .; / 1 ./ : krukL .˝n/ C ho u u d Ä c Cp kukV1H .˝2/ C c kf k 2 L 1C .˝/

Thanks to Proposition 1 we get

2 ˛ 2 2 .; / 2 ./ 1C : krukL2.˝n/ C hoku1 u2kL2. / c Cp c2kukV Ä c kf k 2 (8.9) L 1C .˝/

Taking sufﬁciently small so that ˛c.; Cp/c2 0 and h0 c.; Cp/c2 0, ﬁrstly we can neglect the nonnegative boundary term in (8.9) and deduce (8.7). Neglecting now the terms outside in the left-hand side of (8.9), we deduce (8.8). 88 P. Donato and F. Raimondi

Proposition 3 Under assumptions H1)–H4), let u 2 V be a solution of problem (8.1). Then

. /' ; kf u kL1.˝/ Ä c for every positive ' 2 V with c depending on ˛, ˇ, ,h0,C,c2,C1,C2, k'k 1 p H0 .˝/ and kf kLl.˝/. Proof Let u 2 V be a solution of problem (8.1) and let us choose a nonnegative 1 ' 2 H0 .˝/ as test function in (8.6). Since f , , and ' are nonnegative and the boundary term vanishes, using the Hölder inequality for each term, we have Z Z Z 0 Ä f .u/'dx D B.x; u/rur'dx C u'dx ˝ ˝n ˝ ˇ ' ' : Ä krukL2.˝n/kr kL2.˝n/ C kukL2.˝/k kL2.˝/

From Remark 1 and Proposition 1, thanks to estimates (8.7) and (8.8), we have Z Z B.x; u/rur'dx C u'dx ˝n ˝

1 1 ˇ 1C ' .; ; ; ; / 1C ' : Ä C1kf k 2 kr kL2.˝n/ C C Cp c2 C1 C2 kf k 2 k kL2.˝/ L 1C .˝/ L 1C .˝/

1 The above inequalities imply the result for ' 2 H0 .˝/.Ifnow' D .'1;'2/ 2 V is nonnegative, since is Lipschitz continuous, there exist nonnegative 1 and 2 in 1 .˝/ ' .'1;'2/ . 1 ; 2 /: H0 such that D D j˝1 j˝2 Then Z Z Z Z

0 Ä f .u/'1dx C f .u/'2dx Ä f .u/ 1dx C f .u/ 2dx Ä c: ˝1 ˝2 ˝ ˝

The following proposition estimates the integral of the singular term close to the singular set fu D 0g and it is crucial for the proofs of next results.

Proposition 4 Under assumptions H1)–H4), let u 2 V be a solution of problem (8.1) and ı a ﬁxed positive real number. Then, Z Z

f .u/'dx Ä B.x; u/rur'Zı.u/dx f0ÄuÄıg ˝n

2ı ' 2ı 1 ' ' ; C k kL1.˝/ C khkL . /k 1 C 2kL1. / for every ' 2 V, ' 0, where Zı is deﬁned by (8.4). Proof Let u 2 V be a solution of problem (8.1) and ' 2 V a nonnegative function. Choose, for k >0, Zı.u/Tk.'/ 2 V as test function in (8.6). Using (8.5), we have 8 Singular Elliptic Problems in Two-Component Domains 89 Z Z 1 B.x; u/rurTk.'/Zı.u/dx B.x; u/ruruTk.'/dx ˝n ı f˝n g\fı

C uZı.u/Tk.'/dx C h.u1 u2/ŒZı.u1/Tk.'1/ Zı.u2/Tk.'2/d ˝ Z

D f .u/Zı.u/Tk.'/dx: ˝

From H1) and since ';ı and ˛ are nonnegative, the previous equality implies Z

f .u/Tk.'/dx f0ÄuÄıg Z Z

D f .u/Zı.u/Tk.'/dx Ä f .u/Zı.u/Tk.'/dx f0ÄuÄıg ˝ Z Z

Ä B.x; u/rurTk.'/Zı.u/dx C uZı.u/Tk.'/dx ˝n ˝ Z

C h.u1 u2/ŒZı.u1/Tk.'1/ Zı.u2/Tk.'2/d:

Since u; Zı and Tk are nonnegative, using the Hölder inequality we also have Z Z

uZı.u/Tk.'/dx C h.u1 u2/ŒZı.u1/Tk.'1/ Zı.u2/Tk.'2/d ˝ Z Z

Ä uTk.'/f0ÄuÄ2ıgdx C hŒu1Zı.u1/Tk.'1/ C u2Zı.u2/Tk.'2/d ˝ Z Z 2ı .'/ Œ .' / .' / Ä Tk dx C h u1 f0Äu1Ä2ıgTk 1 C u2 f0Äu2Ä2ıgTk 2 d ˝

2ı .'/ 2ı 1 .' / .' / : Ä kTk kL1.˝/ C khkL . /kTk 1 C Tk 2 kL1. /

For every k >0, we get Z Z

f .u/Tk.'/dx Ä B.x; u/rurTk.'/Zı.u/dx f0ÄuÄıg ˝n

2ı .'/ 2ı 1 .' / .' / : C kTk kL1.˝/ C khkL . /kTk 1 C Tk 2 kL1. /

We can pass to the limit as k tends to inﬁnity in this inequality. Convergence (8.3) and Fatou’s lemma applied on the left-hand side give the result. 90 P. Donato and F. Raimondi

8.4 Main Results

As usual for this kind of problem (see, for instance, [ArEtAl10, CaEtAl11, ChEtAl13, DoEtAl17, GiEtAl16, GiEtAl17] and [GiEtAl09]), in order to study the existence of a solution of problem (8.1), we approximate our problem by a sequence of nonsingular problems and using the a priori estimates stated in the previous section, we pass to the limit. Here, we deﬁne the approximate problem by 8 . . ; / / . . // ˝ ; ˆ div B x um rum C um D Tm f jumj in n ˆ <ˆ um D 0 on @˝; (8.10) ˆ . . ; / / . . ; / / ; ˆ B x um1 rum1 1 D B x um2 rum2 1 on :ˆ .B.x; um1/rum1/ 1 Dh.um1 um2/ on ; where, for every m 2 , the function Tm is the truncation function at level m. The variational formulation associated with the approximate problem (8.10) reads 8 ˆ ˆ Find um 2 V such that ˆ Z Z Z <ˆ B.x; um/rumr'dx C um'dx C h.um1 um2/.'1 '2/d ˝ ˝ ˆ n ˆ Z ˆ : D Tm.f .jumj//'dx; 8' 2 V: ˝

Thanks to this approximation, we avoid the singularity at 0. The existence of a solution of this problem makes use of arguments similar to [Be14] and [BeEtAl17]. Moreover, as in [DoEtAl16] and [DoEtAl17] one can prove that um 0 a. e. in ˝, so that we can write .um/ instead of .jumj/ in the problem.

Theorem 1 Under assumptions H1)–H4), problem (8.6) admits at least a solution.

Proof For m 2 ,letum 2 V be a solution of the approximate problem (8.10). Observe that, since Tm.f .jumj// satisﬁes the same assumptions as the function f .jumj/ for every ﬁxed m 2 , the a priori estimates of Section 8.3 apply uniformly with respect to m. This implies that there exists u 2 V such that, up to a subsequence, 8 ˆ / * ; <ˆ i um u weakly in V ii/ u ! u strongly in L2.˝/ and a.e. in ˝; ˆ m : 2 iii/ um1 um2 ! u1 u2 strongly in L . /:

From now on, the proof follows that of Theorem 5.2 from [DoEtAl16]. 8 Singular Elliptic Problems in Two-Component Domains 91

We also state the two results below, which can be proved following the proofs of the corresponding results from [DoEtAl16]. The ﬁrst one deals with uniqueness and it is stated under the assumption introduced in [Ch09] for a quasilinear problem with Dirichlet boundary conditions.

Theorem 2 Under assumptions H1)–H4), suppose further that is nonincreasing in Œ0; C1Œ and there exists a continuous and nondecreasing function ! W ! , with !.t/>08 t >0, satisfying the following conditions: 8 <ˆ i/ jB.x; t1/ B.x; t2/jÄ!.jt1 t2j/ for a.e. x 2 ˝;8 t1 ¤ t2I Z ˆ y dt : ii/ 8y >0; lim DC1: C x!0 x !.t/

Then problem (8.6) admits a unique solution. Proof We adapt to our case the ideas of [Ch09] (see also [CaEtAl11]). To do that, let u and w be two solutions of problem (8.6). For any >0,setE WD fx 2 ˝ n j .u w/.x/>g and deﬁne 8 Z ˆ s dt < s ; !2. / if F.s/ WD t :ˆ 0 otherwise:

Subtracting the variational formulations of u and v, by similar computations as those done in [Ch09], one can show that r.u w/ C; !. / Ä u w L2.E/ with C independent on . . / . / This inequality can be written as kG u w kV DkrG u w kL2.˝/ Ä C, where 8 Z ˆ s dt < s ; !. / if G.s/ WD t :ˆ 0 otherwise:

This implies that there exists a sequence fkgk2 converging to 0 such that, up to a subsequence, 8 < . /* ; Gk u w G weakly in V : . / 2.˝/ ˝: Gk u w ! G strongly in L and a.e. in 92 P. Donato and F. Raimondi

Accordingly

. /. / . /< ˝ ; lim Gk u w x D G x C1 a.e. in n as k !1I k!0 but from the deﬁnition of G we have also Z .uw/.x/ dt lim G .u w/.x/ D lim DC1 a.e. in E: !0 k !0 !. / k k k t

This involves that jEjD0,as goes to zero, so that u Ä w almost everywhere in ˝ n . Interchanging the roles of u and w, we get u D w almost everywhere in ˝. We state now a boundedness result for a solution of problem (8.6), whose proof makes use of classical ideas from Stampacchia [St65]. In order to get bounded solutions, we need to require a greater summability of the datum f , as done in [ChEtAl13, DoEtAl17, GiEtAl16], and [GiEtAl12]. Hence, instead of H3), we make the following hypothesis:

l N 30 /0f L .˝/; l > : H Ä 2 with 2

Let us mention that, as showed in [DoEtAl17], assumption H30 ) is stronger than H3).

Theorem 3 Under assumptions H1)–H4), suppose further H30 ) holds true. Then, any solution of (8.6) is bounded. Proof For any real number h 1,set

Ah WD fx 2 ˝ n W u.x/>hg and Dh WD fx 2 W u.x/>hg:

Since u 2 V,

lim jAhjD0 and lim @HN1.Dh/ D 0: h!C1 h!C1

C Now, let us choose Gh.u/ WD .u h/ 2 V as test function in (8.6). Since Gh is nondecreasing and 1 1 .u/ Ä < Ä 1 where G .u/ ¤ 0; u h h R R ˛ Œ . /2 . / : we obtain ˝n rGh u dx Ä ˝ fGh u dx From now on, we can use exactly the same arguments of the proof of Theorem 3.2 in [ChEtAl13] (written for ı D 0, g D 0 and with obvious modiﬁcations concerning the boundary term), based on classical results from Stampacchia [St65], to obtain that u belongs to L1.˝/. 8 Singular Elliptic Problems in Two-Component Domains 93

References

[ArEtAl10] Arcoya, D., Boccardo, L., Leonori, T., Porretta, A.: Some elliptic problems with singular natural growth lower order terms. J. Differ. Equ. 249(11), 2771–2795 (2010) [Be14] Beltran, R.J.: Homogenization of a quasilinear elliptic problem in a two component domain with an imperfect interface. Master Thesis, University of the Philippines (2014) [BeEtAl17] Beltran, R.J., Cabarrubias, C., Roque, R.: On the solution of a class of a quasilinear elliptic partial differential equation. Philipp. J. Sci. MS 15-057R (to appear) [CaEtAl11] Cabarrubias, B., Donato, D.: Existence and uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions. Carpathian J. Math. 27, 173–184 (2011) [CaEtAl47] Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Clarendon Press, Oxford (1947) [ChEtAl13] Chourabi, I., Donato, P.: Bounded solutions for a quasilinear singular problem with nonlinear Robin boundary conditions. Differ. Integr. Equ. 26, 975–1008 (2013) [Ch09] Chipot, M.: Elliptic Equations: An Introductory Course: Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel (2009) [DoEtAl17] Donato, P., Giachetti, D.: Existence and homogenization for a singular problem through rough surfaces. SIAM J. Math. Anal. 48(6), 4047–4087 (2016) [DoEtAl16] Donato, P., Monsurrò, S., Raimondi, F.: Existence and uniqueness results for a class of singular elliptic problems in perforated domains. Ricerche Mat. (2016). doi:10.1007/s11587-016-0303-y [GiEtAl09] Giachetti, D., Murat, F.: An elliptic problem with a lower order term having singular behaviour. Boll. Unione Mat. Ital. 2(9), 349–370 (2009) [GiEtAl12] Giachetti, D., Petitta F., Segura de Leon, S.: Elliptic equations having a singular quadratic gradient term and a sign changing datum. Commun. Pure Appl. Anal. 11, 1875–1895 (2012) [GiEtAl16] Giachetti, D., Martínez-Aparicio, J., Murat, F.: An elliptic equation with a mild singularity at u D 0: existence and homogenization. J. Math. Pures Appl. (9), http:// dx.doi.org/10.1016/j.matpur.2016.04.007 (2016) [GiEtAl17] Giachetti, D., Martínez-Aparicio, J., Murat, F.: Deﬁnition, existence, stability and uniqueness of the solution to a similar elliptic problem with a strong singularity at u D 0. Ann. Sc. Norm. Super. Pisa (2017, to appear) [Mo03] Monsurrò, S.: Homogenization of a two-component composite with interfacial thermal barrier. Adv. Math. Sci. Appl. 13, 43–63 (2003) [St65] Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefﬁcients discontinus. Ann. Inst. Fourier (Grenoble) 15(1), 189–258 (1965) Chapter 9 Fredholmness of Nonlocal Singular Integral Operators with Slowly Oscillating Data

G. Fernández-Torres and Yu.I. Karlovich

9.1 Introduction

Let B.X/ denote the Banach algebra of all bounded linear operators acting on a Banach space X,letK.X/ be the closed two-sided ideal of all compact operators in B.X/, and let B .X/ WD B.X/=K.X/. An operator A 2 B.X/ is called Fredholm if its image is closed in X and the spaces ker A and ker A are ﬁnite-dimensional, which is equivalent to the invertibility of the coset A WD A C K.X/ in the Calkin algebra B .X/. We will write A ' B if A B 2 K.X/. Let Cb.C/ denote the C -algebra of all bounded continuous functions on the positive half-line C WD .0; C1/.By[Ka08, p. 232], a function f 2 Cb.C/ is called slowly oscillating (at 0 and 1)if

lim sup fjf .t/ f ./jW t; 2 Œr;2rgD0 for s 2f0; 1g: r!s

The set SO.C/ of all slowly oscillating functions is a C -algebra properly containing C.C/,theC -algebra of all continuous functions on C WD Œ0; C1. Suppose ˛ is an orientation-preserving homeomorphism of C onto itself, which has only two ﬁxed points 0 and 1, and whose restriction to C is a diffeomorphism. 0 We say that ˛ is a slowly oscillating shift if log ˛ 2 SO.C/.Thesetofall slowly oscillating shifts is denoted by SOS.C/.By[KaEtAl11, Lemma 2.2], an orientation-preserving diffeomorphism ˛ of C onto itself belongs to SOS.C/

G. Fernández-Torres () Universidad Nacional Autónoma de México, Ciudad de México, Mexico e-mail: [email protected] Yu.I. Karlovich Universidad Autónoma del Estado de Morelos, Cuernavaca, Morelos, Mexico e-mail: [email protected]

© Springer International Publishing AG 2017 95 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_9 96 G. Fernández-Torres and Yu.I. Karlovich

!.t/ if and only if ˛.t/ D te for t 2 C, where the real-valued functions ! 2 1. / . / !0. / . / 1 !0. / >0 C C and t WD t t belong to SO C and inft2C C t t .The slowly oscillating function !.t/ WD logŒ˛.t/=t is called the exponent function of ˛ 2 SOS.C/. If p 2 Œ1; 1 and ˛ 2 SOS.C/, then the weighted shift operator deﬁned by 0 1=p p U˛f WD .˛ / .f ı ˛/ is an isometric isomorphism of the Lebesgue space L .C/ 1 onto itself. It is clear that U˛ D U˛1 . Given ˛ 2 SOS.C/,letAW denote the Wiener algebra of functional operators X k p A D akU˛ 2 B.L .C// .p 2 Œ1; 1/; k2 P . / 1 < where ak 2 SO C for all k 2 and kAkW WD k2 kakkL .C/ C1. The Cauchy singular integral operator S D SC and the operator R, deﬁned by Z Z 1 f ./ 1 f ./ .S f /.t/ WD d; .Rf /.t/ WD d for t 2 ; C C i C t i C C t (9.1) with integrals understood in the principal value sense, are bounded on the spaces p L .C/, 1

Let M.A/ be the maximal ideal space of a unital commutative Banach algebra A. Identifying points t 2 C with the evaluation functionals t.f / D f .t/ for f 2 C.C/, we get M.C.C// D C. Consider the ﬁbers ˚ « . . // . . // Ms SO C WD 2 M SO C W jC.C/ D s of the maximal ideal space M.SO.C// over points s 2f0; 1g.By[Ka08, Proposition 2.1], the set

WD M0.SO.C// [ M1.SO.C// coincides with closSO C n C, where closSO C is the weak-star closure of C in the dual space of SO.C/. Then M.SO.C// D [ C. In what follows we write a./ WD .a/ for every a 2 SO.C/ and every 2 .Letp 2 .1; 1/ and let

sp.x/ WD cothŒ .x C i=p/; rp.x/ WD 1= sinhŒ .x C i=p/; x 2 : (9.3) 9 Fredholmness of Nonlocal Singular Integral Operators 97

With the operator N we associate the function n deﬁned on C by Â X Ã Â X Ã . ; / C. / ik!.t/x . / . / ik!.t/x . /; n t x D ak t e pC x C ak t e p x (9.4) k2 k2 where ! 2 SO.C/ is the exponent function of ˛ and p˙.x/ WD .1 ˙ sp.x//=2. As n.; x/ 2 SO.C/ for every x 2 , taking the Gelfand transforms of n.; x/,we extend these functions to all M.SO.C//. Hence, for all .; x/ 2 . [ C/ , Â X Ã Â X Ã .; / C./ ik!./x . / ./ ik!./x . /: n x D ak e pC x C ak e p x (9.5) k2 k2

For every 2 , the function n.; / is semi-almost periodic [Sa77]. Moreover, n.; / 2 SAPp, where SAPp is the smallest closed subalgebra of the Banach p algebra Mp./ of Fourier multipliers on L ./ that contains Cp./ and APp (see [BoEtAl02, p. 372]), Cp./ is the closure in Mp./ of the set C./\V./ [Du79], M ./ and APpPis the closure in p of the set of all almost periodic polynomials . / ix ¼ ˝ p x D 2˝ ce , where c 2 , 2 , and is a finite subset of . For p 2 .1; 1/, sufficient Fredholm conditions for the operator (9.2) with bino- ˙ ˙ p mial functional operators A˙ D a0 I a1 U˛ on the space L .C/ were obtained in [KaEtAl11]. The necessity of these conditions was proved in [KaEtAl11a]. One- sided invertibility criteria for such operators A˙ were established in [KaEtAl16]. Recently [FeKa17] we obtained criteria for the two-sided and one-sided invertibility of Wiener type functional operators A 2 AW on every Lebesgue space p L .C/, p 2 Œ1; 1. In the present paper, applying the Allan-Douglas local principle, results on Mellin pseudodifferential operators with symbols of limiting smoothness that follow from [Ka06], techniques of limit operators and invertibility criteria from [FeKa17] for operators A 2 AW , we establish a Fredholm criterion for p the operator (9.2) on the Lebesgue space L .C/ for any p 2 .1; 1/, where we reduce the smoothness of the slowly oscillating shift ˛ (cf. [Ka08]). 1< < ˙ . / Theorem 1 (Main Result) Suppose p 1,ak 2 SO C for all k 2 , and ˛ 2 SOS.C/. Then the operator N given by (9.2) is Fredholm on the space p L .C/ if and only if the following two conditions hold: P ˙ k A p. / (i) the functional operators A˙ D k2 ak U˛ 2 W are invertible on L C ; (ii) the function n defined by (9.3)–(9.5) possesses the property

inffjn.; x/jW x 2 g >0 for every 2 : (9.6)

Theorem 1 remains valid if to replace (9.6) in condition (ii) by

n.; x/ ¤ 0 for all .; x/ 2 : (9.7)

Indeed, the necessity follows from Theorem 1, because (9.6) implies (9.7). The sufﬁciency will be proved in Section 9.4 by using, in fact, (9.7) instead of (9.6). 98 G. Fernández-Torres and Yu.I. Karlovich

9.2 Invertibility Criteria for Wiener Type Functional Operators

Theorem 2 [FeKa17, Theorem 7.4] The Wiener algebra AW is inverse closed in the p Banach algebra B.L .C// for every p 2 Œ1; 1, that is, every operator A 2 AW p invertible in B.L .C// is also invertible in AW . Let ˛ 2 SOS.C/. Suppose ˛0.t/ WD t and ˛n.t/ WD ˛Œ˛n1.t/ for all n 2 and all t 2 C.Given 2 C, we put ˙ WD limn!˙1 ˛n./ 2f0; 1g.Fix 2 C, and let be a semi-segment of C with endpoints and ˛./, where 2 and ˛./ … . Given p 2 Œ1; 1, we consider the Banach space lp D lp./ with usual norm. P Œ1; k Theorem 3 [FeKa17, Theorems 3.1, 3.2] If p 2 1 and A D k2 akU˛ 2 p AW B.L .C//, then the function A, deﬁned by

A.t/ D .ajiŒ˛i.t//i;j2 I for all t 2 C; (9.8)

p is a bounded continuous B.l /-valued operator function on C, and

A. / : kAkB.Lp.C// D max k t kB.lp/ ÄkAkW t2C

1 p p Moreover, Aj I D A 2 B.L . ; l //, where is the isometric isomorphism ˚ « p. / p. ; p/; p; . n /. / : W L C ! L l f 7! I W ! l t 7! U˛f t n2

The operators A.t/ deﬁned for all t 2 C are called discrete or band-dominated operators. Applying Theorem 3, we arrive at the ﬁrst invertibility criterion (see [FeKa17, Theorem 3.4]).

Theorem 4 For p 2 Œ1; 1, a functional operator A 2 AW is invertible on the p space L .C/ if and only if for all t 2 (equivalently, for all t 2 C) the discrete operators A.t/ given by (9.8) are invertible on the space lp. B. / U 1 Let X be a Banach space, let X be its dual space, A 2 X and let DfUngnD1 be a sequence of isometries. If the strong limits

1 1 AU WD s-lim.U AUn/ in B.X/; AU WD s-lim.U AUn/ in B.X / n!1 n n!1 n

exist, then always .AU/ D AU , and we will refer to the operator AU as the limit operator for the operator A with respect to the sequence U (see, e.g., [BoEtAl00, RaEtAl04]). We use the following variation of [GoFe74, Chap. 3, Lemma 1.1]. Corollary 1 If an operator A 2 B.X/ is invertible in B.X/ and the strong limits AU 2 B.X/ and AU 2 B.X / exist with respect to a sequence U of isometries in B.X/, then the limit operator AU is also invertible in B.X/. Applying [FeKa17, Lemma 4.3] and Corollary 1, we infer the following. 9 Fredholmness of Nonlocal Singular Integral Operators 99

TheoremP 5 [FeKa17, Theorem 5.2] Let p 2 .1; 1/. If the operator k A . / ˛ . / A D k2 akU˛ 2 W with coefﬁcients ak 2 SO C and a shift 2 SOS C p. / is invertibleP on the space L C , then for all 2 the limit operators ./ k A A WD k2 ak U˛ 2 W are also invertible on this space, and hence X k A .z/ WD ak./z ¤ 0 for all 2 and all z 2 : k2 By Theorem 5 and [FeKa17, Corollary 5.4], we get the following. P k A Corollary 2 If the functional operator A D k2 akU˛ 2 W is invertible on the p Lebesgue space L .C/ with p 2 .1; 1/, then the Cauchy indices

1 ind A ./ WD .2 / farg A .z/gz2; where farg A .z/gz2 denotes the increment of any continuous branch of arg A .z/ . . // as z traces the unit circle counter-clockwise, coincide for all 2 MC SO C and, respectively, for all 2 M .SO.C//, which uniquely deﬁnes the numbers . / . . // 0; : N˙ WD ind A for all 2 M˙ SO C and ˙ 2f 1g (9.9) Given n 2 , we consider the projections ( ( ˚ « 0 < ; 1 ; ˙ ˙ B. p/; C if s n if s Än Pn D diag Ps;n I 2 l Ps;n D Ps;n D s2 1 if s n; 0 if s > n:

Theorem 5, Corollary 2 and [FeKa17, Theorem 6.6] imply the following. P k A Theorem 6 If the operator A D k2 akU˛ 2 W is invertible on the space p L .C/ with p 2 .1; 1/, then there is a number n0 2 such that for all n n0 and all t 2 the operators " # A. / A. / C Pn t PnN Pn t PnCNC D ; .t/ WD (9.10) n 1 CA. / CA. / C Pn t PnN Pn t PnCNC

: : p C p p C p are invertible from the space PnN l C PnCNC l onto the space Pn l C Pn l , and h i D . / 0A. / 0 0A. / 0A. / C n;0 t WD Pn t PnN;nCNC Pn t PnN Pn t PnCNC " # " # 1 C 0 P A.t/P P A.t/P P A.t/P ; n nN n nCNC n nN nCNC ; CA. / CA. / C CA. / 0 Pn t PnN Pn t PnCNC Pn t PnN;nCNC (9.11)

0 p 0 p act from the space PnN;nCNC l to the space Pnl , where the numbers N˙ are given 0 C 0 C by (9.9),PnN;nCNC D I PnN PnCNC and Pn D I Pn Pn . 100 G. Fernández-Torres and Yu.I. Karlovich

Applying Theorems 4, 6 and [FeKa17, Theorem 6.9], and identifying the operators Dn;0.t/ with their matrix representations, we arrive at the second invertibility criterion for Wiener type functional operators A 2 AW . P k Theorem 7 [FeKa17, Theorem 8.1] The functional operator A D k2 akU˛ 2 AW with coefﬁcients ak 2 SO.C/ and a shift ˛ 2 SOS.C/ is invertible on the p space L .C/ for p 2 Œ1; 1 if and only if P . / ./ k 0 (i) A z WD k2 ak z ¤ for every 2 and every z 2 ; . / . . // (ii) N D NC, where N˙ WD ind A for every 2 M˙ SO C ; (iii) there exists an n0 2 such that for every t 2 and every n > n0 the operator Dn;1.t/ given by (9.10) is invertible and det Dn;0.t/ ¤ 0, where the .2n 1/ .2n 1/ matrices Dn;0.t/ are given by (9.11).

9.3 Mellin Pseudodifferential Operators

Consider the Mellin transform M and its inverse M1 deﬁned by Z 2 2 ix dt M W L .C; d/ ! L ./; .Mf /.x/ WD f .t/t ; C t Z 1 1 2 2 1 ix M W L ./ ! L .C; d/; .M g/.t/ WD g.x/t dx: 2

p For p 2 .1; 1/,letMp./ be the Banach algebra of Fourier multipliers on L ./. 1 If a 2 Mp./, then the Mellin convolution operator Co.a/ WD M aM is bounded p on the Banach space L .C; d/, where d.t/ D dt=t is the (normalized) invariant measure on C. Consider the isometric isomorphism

p p 1=p ˚ W L .C/ ! L .C; d/; .˚f /.t/ WD t f .t/; t 2 C: (9.12)

; ˚ 1 The operators SC R given by (9.1) are represented in the form SC D 1 Co.sp/˚ and R D ˚ Co.rp/˚, where the functions sp; rp 2 Cp./ are given by (9.3). Let V./ be the Banach algebraR of all absolutely continuous functions a W ! 0 ¼ of ﬁnite total variation V.a/ D ja .x/j dx, with the norm kakV WD kakL1./ C V.a/. By Stechkin’s inequality [Du79, Theorem 2.11], V./ Mp./. Let Cb.C; V.// be the Banach algebra of all bounded continuous V./- a. ; / a. ; / valued functions on C with the norm k kCb.C;V.// D supt2C k t kV , and 1 let C0 .C/ be the set of all inﬁnitely differentiable functions of compact support on C. Mellin pseudodifferential operators generalize Mellin convolution operators. By [Ka06, Theorem 3.1] and [KaEtAl14, Theorem 3.1], we have the following.

Theorem 8 If a 2 Cb.C; V.//, then the Mellin pseudodifferential operator 1 Op.a/, deﬁned for f 2 C0 .C/ by the iterated integral 9 Fredholmness of Nonlocal Singular Integral Operators 101

Z Z Á 1 t ix d Œ .a/f .t/ dx a.t; x/ f ./ for t Op D 2 2 C C

p extends to a bounded linear operator on the space L .C; d/ and there is a number Cp 2 .0; 1/ depending only on p such that

.a/ p a : kOp kB.L .C;d// Ä Cpk kCb.C;V.//

Consider the Banach subalgebra SO.C; V.// of the algebra Cb.C; V.// consisting of all V./-valued functions a on C that slowly oscillate at 0 and 1, that is,

lim max ka.t; / a.; /kL1./ D 0; s 2f0; 1g: r!s t;2Œr;2r

Let E.C; V.// be the Banach algebra of all V./-valued functions a in the h . ; .// 0 a. ; / a . ; / 0 algebra SO C V such that limjhj! supt2C t t V D , where h a .t; x/ WD a.t; x C h/ for all .t; x/ 2 C . Theorem 3.3, Lemma 3.4 in [KaEtAl14](seealso[Ka06]) imply the following.

Theorem 9 If a; b; c; d 2 E.C; V.//, a depends only on the ﬁrst variable and c depends only on the second variable, then

Op.b/Op.d/ ' Op.bd/; Op.a/Op.b/Op.c/ D Op.abc/:

The following result plays a crucial role in the Fredholm study of operator (9.2).

Lemma 1 [KaEtAl14, Lemma 4.5] Suppose ˛ 2 SOS.C/, ! is its exponent p function, U˛ is the associated isometric shift operator on L .C/, and R is given by (9.1). Then the operator U˛R can be realized as the Mellin pseudodifferential 1 operator up to a compact operator: U˛R ' ˚ Op.c/˚, where the function c, given i!.t/x by c.t; x/ WD e rp.x/ for .t; x/ 2 C , belongs to the algebra E.C; V.//.

9.4 Fredholmness of the Operator N

Let A be a Banach algebra and S be a subset of A. We denote by algAS the smallest closed subalgebra of A containing S and by idAS the smallest closed two-sided A S B B. p. // C ; ideal of containing . Put WD L C , WD algBfI Sg, where S D SC . Obviously, the algebra C is commutative. Fix ˛ 2 SOS.C/ and consider the Banach algebra F˛ WD algB.fSg[AW / of singular integral operators with shift. Following [KaEtAl11, Section 6.3], we also consider the unital Banach algebras:

Z WD algBfI; S; cR; K W c 2 SO.C/; K 2 Kg; p WD fA 2 B.L .C// W AC CA 2 K for all C 2 Zg; 102 G. Fernández-Torres and Yu.I. Karlovich where the operator R 2 C with ﬁxed singularities at 0 and 1 is given by (9.1), p K WD K.L .C//, and is called the algebra of operators of local type relative to Z. 1 1 Since S D ˚ Op.sp/˚ and cR D ˚ Op.c0/˚ for every c 2 SO.C/, where the functions sp.t; x/ WD sp.x/ and c0.t; x/ WD c.t/rp.x/, deﬁned for .t; x/ 2 C , belong to the algebra E.C; V.// by [KaEtAl11, Lemma 7.1], we infer from 1 Theorem 9 that ˚ Op.b/˚ 2 if b 2 E.C; V.//. We deduce the following result from [KaEtAl11, Theorem 6.8].

Lemma 2 If ˛ 2 SOS.C/,A2 AW and B 2 C, then AB ' BA. By Lemma 2 and [KaEtAl11, Theorem 6.8], we get

K Z F˛ : (9.13)

Hence, the quotient algebras Z WD Z=K and WD =K are well deﬁned and Z lies in the center of .By[KaEtAl11, Theorem 6.11], the maximal ideal space M.Z / of the commutative Banach algebra Z is homeomorphic to the set f˙1g[ . / J J .; / .Let ˙1 and ;x for x 2 denote the closed two-sided ideals of the Banach algebra generated, respectively, by the maximal ideals ˚ « ˚ «

. / ;. / . / ; Z . /b.; / 0 I˙1 WD idZ P gR W g 2 SO C I;x WD Z 2 W Z x D of the algebra Z , where .Z /b.; x/ is the Gelfand transform of Z 2 Z , and let =J =J ˙1 WD ˙1 and ;x WD ;x be the corresponding quotient algebras. p Obviously, an operator T 2 is Fredholm on the space L .C/ if and only if the coset T WD A C K is invertible in the quotient Banach algebra B . Applying the Allan-Douglas local principle (see, e.g., [BoSi06, Theorem 1.35(a)]) to the Banach algebra and its central subalgebra Z , we obtain the following. p Theorem 10 An operator T 2 is Fredholm on the space L .C/ if and only if J the cosets T C ˙1 are invertible in the quotient algebras ˙1, respectively, and .; / J for every x 2 the coset T C ;x is invertible in the quotient algebra ;x. As in [KaEtAl11, Theorem 8.1], Theorem 2 and AW imply the following. ˙ . / ˛ . / Theorem 11 Let ak 2 SO C forP all k 2 , 2 SOS C , and N be given ˙ k A by (9.2). If some operator A˙ D k2 ak U˛ 2 W is invertible on the space p. / J L C , then the coset N C ˙1 is invertible in the quotient algebra ˙1, respectively. Slightly modifying the proof of [KaEtAl11, Lemma 8.3], we get the following. Lemma 3 Suppose ˛ is a slowly oscillating shift, ! is its exponent function, and p U˛ is the associated isometric shift operator on L .C/.If.; x/ 2 , then

. 2/ i!./x. . //2 J : U˛R e rp x I 2 ;x 9 Fredholmness of Nonlocal Singular Integral Operators 103

˙ . / ˛ . / Theorem 12 Suppose ak 2 SO C for all k 2 , 2 SOS C and N is given by (9.2).Ifn.; x/ ¤ 0 for some .; x/ 2 , where the function n is deﬁned J by (9.3)–(9.5), then the coset N C ;x is invertible in the quotient algebra ;x. 2 2 Fix .; x/ 2 and consider the operators H˙ WD p˙.x/Œrp.x/ R . Since

. / ;.C / . C./ / ;. / . ./ / J P˙ H˙ ak HC ak HC ak H ak H 2 I;x ;x

k k and .U˛H˙/ D .H˙U˛/ , we infer by analogy with [KaEtAl11, Theorem 8.4] that X J C./ k ./ k J : N C ;x D ak U˛HC C ak U˛H C ;x (9.14) k2

. k / ik!./x . / J By deﬁnition of H˙ and Lemma 3, U˛H˙ e p˙ x I 2 ;x. Combining J .; / J .; / this with (9.14), we deduce that N C ;x D n x I C ;x, where n x is given .; / 0 .1= .; // J by (9.3)–(9.5). If n x ¤ , then n x I C ;x is the inverse of the coset J N C ;x in the quotient algebra ;x, which proves Theorem 12. Since N 2 by (9.13), the proof of sufﬁciency in Theorem 1 follows now from Theorems 10, 11, and 12. It remains to prove the necessity. Let X be a Banach space, A 2 B.X/ and let jAjC WD inffkAxkX WkxkX D 1g be a lower norm of A. Then [Ku99, Theorem 1.3.2] implies the following.

Theorem 13 An operator A 2 B.X/ is invertible if and only if jAjC >0and 1 jA jC >0. If A is invertible, then jAjC D 1=kA kB.X/. ˙ . / ˛ . / Theorem 14 PLet ak 2 SO C for all k 2 , 2 SOS C ,ND ACPC CAP, ˙ k A . /=2 where A˙ D k2 ak U˛ 2 W and P˙ D I ˙S . If the operator N is Fredholm p on the space L ./ with p 2 .1; 1/, then the functional operators A˙ are invertible on this space. To prove Theorem 14, which gives the necessity of condition (i) in Theorem 1, we apply ideas of [Ka89, KaSi02] (for earlier approaches see [KaKr84]). The proof is based on the fact that if, for example, the functional operators AC 2 AW in (9.2) p is not invertible, then either there exists a sequence of functions fn 2 L .C/ of norm 1 such that the sequences fACfng and fPCPfng converge, but the sequence fPCfng does not contain convergent subsequences, or there exists a sequence of q. /.1= 1= 1/ 1 functions gn 2 L C p C q D of norm such that the sequences fACgng . / and f PCP gng converge, but the sequence fPCgng does not contain convergent subsequences (see [KaSi04, p. 57] and [Ka08, p. 249]). By Lemma 2, the commutators A˙P PA˙ are compact for P 2fP˙g.Ifthe operator AC is not invertible, then by Theorem 4 there exists a point 2 C such that the discrete operator AC./ given by (9.8) is not invertible on the space p l . We then deduce from Theorem 13 that one of the lower norms jAC./jC or j.AC.// jC is equal to zero. Let for deﬁniteness jAC./jC D 0. Then we p construct a required sequence of functions fn 2 L .C/ of norm 1 by following the main lines of the proof of [KaSi04, Theorem 5.2]. A sequence of functions q gn 2 L .C/ of norm 1 in the case j.AC.// jC D 0 is constructed analogously. 104 G. Fernández-Torres and Yu.I. Karlovich

To ﬁnish the proof of Theorem 1, we partially follow [KaEtAl11a]. .1; / ˙ . / ˛ . / Theorem 15 Suppose p 2 1 ,ak 2 SO C for all k 2 , 2 SOS C , the operator N is given by (9.2), and the function n is given by (9.3)–(9.5).Ifthe p operator N is Fredholm on the space L .C/, then infx2 jn.; x/j >0for every 2 . To prove Theorem 15, for every k 2 C, we take the isometric dilation operator p 1=p Vk 2 B.L .C// given by .Vkf /.t/ WD k f .t=k/, t 2 C.Fixs 2f0; 1g.As in [KaEtAl11a, Lemma 4.5], for every 2 Ms.SO.C// there exists a sequence 1 h Dfh;ngnD1 C such that limn!1 h;n D s and there exists the limit operator Â X Ã Â X Ã . 1 / C./ k ./ k ; N WD s-lim V NVh; D a U˛ PC C a U˛ P (9.15) n!1 h;n n k k k2 k2

!./ where ˛ .t/ D e t for t 2 C.IfN is Fredholm, then by Corollary 1 the limit 1 operator N givenby(9.15) is invertible. On the other hand, N D ˚ Co.n.; //˚, where n.; / 2 SAPp and ˚ are given by (9.5) and (9.12). Hence, Co.n.; // is p invertible on the space L .C; d/. Then, by [BoEtAl02, Proposition 19.4], for every 2 ,infx2 jn.; x/j >0, which gives condition (ii) of Theorem 1. Finally, combining Theorems 14 and 15, we obtain the necessity of conditions (i)–(ii) in Theorem 1. This completes the proof of Theorem 1.

References

[BoKa97] Böttcher, A., Karlovich, Yu.I.: Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Birkhäuser, Basel (1997) [BoSi06] Böttcher, A., Silbermann, B.: Analysis of Toeplitz Operators. Springer, Berlin (2006) [BoEtAl00] Böttcher, A., Karlovich, Yu.I., Rabinovich, V.S.: The method of limit operators for one-dimensional singular integrals with slowly oscillating data. J. Oper. Theory 43, 171–198 (2000) [BoEtAl02] Böttcher, A., Karlovich, Yu.I., Spitkovsky, I.M.: Convolution Operators and Factor- ization of Almost Periodic Matrix Functions. Birkhäuser, Basel (2002) [Du79] Duduchava, R.: Integral Equations with Fixed Singularities. Teubner Verlagsge- sellschaft, Leipzig (1979) [FeKa17] Fernández-Torres, G., Karlovich, Yu.I.: Two-sided and one-sided invertibility of the Wiener type functional operators with a shift and slowly oscillating data. Banach J. Math. Anal., advance publication, 3 May 2017. doi:10.1215/17358787-2017-0006. http://projecteuclid.org/euclid.bjma/1493776976 [GoFe74] Gohberg, I.C., Fel’dman, I.A.: Convolution Equations and Projection Methods for Their Solution. Translations of Mathematical Monographs, vol. 41. American Mathematical Society, Providence, RI (1974) [KaEtAl11] Karlovich, A.Yu., Karlovich Yu.I., Lebre, A.B.: Sufﬁcient conditions for Fredholm- ness of singular integral operators with shifts and slowly oscillating data. Integr. Equ. Oper. Theory 70, 451–483 (2011) [KaEtAl11a] Karlovich, A.Yu., Karlovich Yu.I., Lebre, A.B.: Necessary conditions for Fredholm- ness of singular integral operators with shifts and slowly oscillating data. Integr. Equ. Oper. Theory 71, 29–53 (2011) 9 Fredholmness of Nonlocal Singular Integral Operators 105

[KaEtAl14] Karlovich, A.Yu., Karlovich Yu.I., Lebre, A.B.: Fredholmness and index of simplest singular integral operators with two slowly oscillating shifts. Oper. Matrices 8, 935–955 (2014) [KaEtAl16] Karlovich, A.Yu., Karlovich Yu.I., Lebre, A.B.: One-sided invertibility criteria for binomial functional operators with shift and slowly oscillating data. Mediterr. J. Math. 13, 4413–4435 (2016) [Ka89] Karlovich, Yu.I.: On algebras of singular integral operators with discrete groups of shifts in Lp-spaces. Sov. Math. Dokl. 39, 48–53 (1989) [Ka06] Karlovich, Yu.I.: An algebra of pseudodifferential operators with slowly oscillating symbols. Proc. Lond. Math. Soc. 92, 713–761 (2006) [Ka08] Karlovich, Yu.I.: Nonlocal singular integral operators with slowly oscillating data. In: Bastos, M.A., et al. (eds.) Operator Algebras, Operator Theory and Applications. Operator Theory: Advances and Applications, vol. 181, pp. 229–261. Birkhäuser, Basel (2008) [KaKr84] Karlovich, Yu.I., Kravchenko, V.G.: An algebra of singular integral operators with piecewise-continuous coefﬁcients and piecewise-smooth shift on a composite contour. Math. USSR Izvestiya 23, 307–352 (1984) [KaSi02] Karlovich, Yu.I., Silbermann, B.: Local method for nonlocal operators on Banach spaces. In: Böttcher, A., et al. (eds.) Toeplitz Matrices and Singular Integral Equations. Operator Theory: Advances and Applications, vol. 135, pp. 235–247. Birkhäuser, Basel (2002) [KaSi04] Karlovich, Yu.I., Silbermann, B.: Fredholmness of singular integral operators with discrete subexponential groups of shifts on Lebesgue spaces. Math. Nachr. 272, 55–94 (2004) [Ku99] Kurbatov, V.G.: Functional-Differential Operators and Equations. Kluwer Academic Publishers, Dordrecht (1999) [RaEtAl04] Rabinovich, V.S., Roch, S., Silbermann, B.: Limit Operators and Their Applications in Operator Theory. Birkhäuser, Basel (2004) [RoEtAl11] Roch, S., Santos, P.A., Silbermann, B.: Non-Commutative Gelfand Theories. A Tool- kit for Operator Theorists and Numerical Analysts. Springer, Berlin (2011) [Sa77] Sarason, D.: Toeplitz operators with semi-almost periodic symbols. Duke Math. J. 44, 357–364 (1977) Chapter 10 Multidimensional Time Fractional Diffusion Equation

M. Ferreira and N. Vieira

10.1 Introduction

Time fractional diffusion equations are obtained from the standard diffusion and wave equations by replacing the time derivative by a fractional derivative of order ˇ 2 0; 2. When ˇ D 0 we have the Helmholtz equation, which is associated with localized diffusion. The subdiffusion case corresponds to the case when 0< ˇ<1. When ˇ D 1 we obtain the ordinary diffusion. The superdiffusion regime happens when 1<ˇ<2.Ifˇ D 2, we are dealing with the wave equation (sometimes designated as ballistic diffusion) (see [Po07]). Some applications of the time fractional diffusion equation can be found in the study of Brownian motion [MeKl00] and in the theory of thermoelasticity [LoSh67]. The theory of generalized thermoelasticity appeared due to the use of fractional derivatives with respect to time or space to describe anomalous diffusion processes. The time fractional diffusion (or heat conduction) equation

M. Ferreira () School of Technology and Management, Polytechnic Institute of Leiria, P-2411-901, Leiria, Portugal CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal e-mail: [email protected] N. Vieira CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal e-mail: [email protected]

dˇu D c u;0<ˇÄ 2 (10.1) dtˇ

dˇ u where dtˇ is a fractional derivative usually considered in the Riemann–Liouville or in the Caputo sense, describes the time non-local dependence between the ﬂux vectors and corresponding gradients with “long-tale” power kernel. Equation (10.1) is also important in the study of Brownian motion. At the level of individual particle motion the classical diffusion corresponds to the Brownian motion which is characterized by a mean-squared displacement increasing linearly with time hx2i at: In the case of fractional Brownian motion (anomalous diffusion) the mean-squared displacement increases with the power-law time, that is, hx2i atˇ; ˇ ¤ 1. Therefore, the continuous time random walk theory can be generalized to include variable jump lengths and waiting times between successive jumps, giving rise to subdiffusion processes (0<ˇ<1) and superdiffusion processes (1<ˇ < 2) (see [MeEtAl98]). In the one-dimensional case, fundamental solutions and solutions for Cauchy problems related to (10.1) have been constructed and studied comprehensively in several works (see, for example, [Fu90, KiEtAl06, Lu13b, MaPa03, ScWy89, Wy86]) and the references therein indicated. For the multidimensional case there are some works in this direction (see, e.g., [Ha02a, Ha02b, KiEtAl06, Lu13a, ScWy89]). However, in these works it was not obtained a series representation for the fundamental solution in an arbitrary dimension. A representation of the fundamental solution in the form of an absolute convergent series is important since it enables to use these functions in approximation problems. In this paper we obtain explicitly integral and series representations for the fundamental solution of the time fractional diffusion equation, for an arbitrary dimension. The series representations obtained depend on the parity of the dimension. As an application of our results we study also diffusion and stress in the axially symmetric case for plane deformation. Integral representations for the solution of this problem were obtained by Povstenko in [Po07] using Fourier–Laplace techniques. The structure of the paper reads as follows: in the preliminaries section we recall some basic concepts about fractional calculus and special functions. In Section 10.3 we construct integral and series representations for the fundamental solution of the time fractional diffusion equation. In Section 10.4 we compute the fractional moments of arbitrary order >0and in Section 10.5 we present some plots of the fundamental solution for n D 1 and n D 2: Finally, in Section 10.6 we study the problem of diffusion and stress in the axially symmetric case for plane deformation.

10.2 Preliminaries

For a locally integrable function f on n, the multidimensional Fourier transform of f is the function deﬁned by the integral Z .F f /./ Dbf .k/ D eix f .x/ dx; n 10 Multidimensional Time Fractional Diffusion Equation 109

n n where x; 2 Pand x denotes the usual inner product in . For the Laplace n @2 cf ./ 2 bf ./ operator D iD1 xi we have Djj . Finally, we recall the C ˇ definition of the Caputo fractional derivative @t of order ˇ>0(see [KiEtAl06]) Z Á t m ˇ 1 1 @ f C@ . ; / . ; / ; t f x t D ˇ 1 x w dw .m ˇ/ 0 .t w/ mC @wm where m D Œˇ C 1; t >0;and Œˇ means the integer part of ˇ.Forˇ D m 2 ,the Caputo fractional derivative coincides with the standard derivative of order m. Some special functions naturally arise in the study of fractional calculus. For our work we need the one-parametric Mittag–Leffler function E˛ (see [GoEtAl14]), which is defined in terms of the power series by

X1 n z C E˛.z/ D ; z 2 ¼; <.˛/ 2 : .˛n C 1/ nD0

Another special function with an important role in fractional calculus is the Wright function W˛;ˇ (see [GoEtAl99]) which is deﬁned as the convergent series

C1X zn W˛;ˇ.z/ D ;˛>1; ˇ 2 ¼: .˛n C ˇ/ nŠ nD0

The Wright function is related with the Bessel function of ﬁrst kind with index by the formula

C1 Á Á Â Ã X .1/n z 2kC z z2 J .z/ D D W1; 1 : .n C C 1/ kŠ 2 2 C 4 nD0

The fundamental solution obtained in this paper can be represented in terms of the m;n Fox H-function Hp;q , which is deﬁned by a Mellin-Barnes type integral in the form (see [KiSa04])

" # Z Q Q m . ˇ / n .1 ˛ / .a1;˛1/;:::;.a ;˛/ 1 jD1 bj C js iD1 ai is Hm;n p p D Q Q zs ds; p;q z . ;ˇ/;:::;. ;ˇ/ 2 p . ˛ / q .1 ˇ / b1 1 bq q i L iDnC1 ai C is jDmC1 bj js

C where ai; bj 2 ¼; and ˛i;ˇj 2 ; for i D 1;:::;p and j D 1;:::;q; and L is a suitable contour in the complex plane separating the poles of the two factors in the numerator (see [KiSa04]). 110 M. Ferreira and N. Vieira

10.3 Fundamental Solution of the Multidimensional Time Fractional Diffusion-Wave Equation

In [FeVi17] we have considered the following Cauchy problem 8 Á ˆ C@ˇ 2 ˇ. ; / 0 ˆ t c Gn x t D ˆ Yn < ˇ G .x;0/D ı.xi/; 0 < ˇ Ä 2 ˆ n ˆ iD1 ˆ ˇ ˆ @Gn : .x;0/D 0; 1 < ˇ Ä 2 @t where for 0<ˇÄ 1 we only consider the ﬁrst initial condition and for 1<ˇÄ 2 we need to consider the two initial conditions, and ı is the delta Dirac function. Using the Fourier transform with respect to space we obtain the following unique solution in Fourier domain given in terms of the Mittag-Lefﬂer function (see [FeVi17]) bˇ.; / 2 2 ˇ ;ˇ0; 2: Gn t D Eˇ c j j t 2

Applying the inverse Fourier transform we get the following integral representation for the fundamental solution: Z 1 ˇ. ; / ix 2 2 ˇ ; n; >0: Gn x t D n e Eˇ c j j t d x 2 t (10.2) .2 / n 2 2 ˇ Since Eˇ c jj t is a radial function in ; (10.2) can be rewritten as

n Z 1 2 C1 ˇ jxj n 2 2 ˇ . ; / 2 n . / : Gn x t D n Eˇ c t J 2 1 jxj d (10.3) .2 / 2 0

n n 1: where J 2 1 is the Bessel function of ﬁrst kind with index 2 Using the Mellin transform and the Mellin convolutionn o theoremn it is possible Áo to rewrite (10.3)as ˇ 1 a Mellin convolution, i.e., M G .s/ D M .f M g/ .s/, where g./ D n jxj 2 2 ˇ 1 1 ./ n Eˇ c t and f D n n C1 J 1 . Applying the inverse Mellin .2 / 2 jxjn 2 2 transform we obtain a representation of (10.3) as a Mellin-Barnes integral and, consequently, as a Fox H-function (for more details see [FeVi17]): ! ˇ s Z s ns 1 1 Ci1 1 2ct2 Gˇ.x; t/ D 2 Á 2 ds n n n ˇ 2 2 jxj 2 i i1 1 s jxj 2

(10.4) 10 Multidimensional Time Fractional Diffusion Equation 111 " # ˇ 0; 1 ; 1 n ; 1 1 2 2 2 2 2 0;2 ct Á : D n H2;1 ˇ (10.5) n 0; 2 2 jxj jxj 2

ˇ Remark 1 The representation (10.5)ofGn in terms of the H-function coincides with the one given in [KiEtAl06, Sec. 6.2, (6.3.33)] under a suitable change of variables in the Mellin–Barnes integral. The integral (10.4) can be computed explicitly applying the Residue Theorem where the contour of integration must be transformed to the loop LC1 (see [KiSa04 ]) 1 s starting and ending at infinity and encircling all the poles of the functions 2 ns . / and 2 in the numerator. Since the gamma function s has simple poles at ; ; 1 s 2 2; ; s Dn with n 2 0 we have that 2 has poles at s D k C with k 2 0 ns 2 ; : and 2 has poles at s D k C n with k 2 0 Therefore, in the odd case we have two non-coincident sequences of simple poles, s D 2k C 2; k 2 0; and s D 2k C n; k 2 0;while in the even case we have a finite sequence of simple 1 s 2 2; 0; 1; : : : ; n 2; poles coming from 2 at the points s D k C for k D 2 1 s ns and an infinite sequence of double poles coming from 2 2 at the 2 2; n 1: points s D k C for k 2 Applying the Residue Theorem and after some manipulations (see [FeVi17]) we obtain the following result. Theorem 1 For n odd and 0<ˇÄ 2 the FS of the time fractional diffusion-wave operator is given by

n3 Â Ã 1 X2 1 k C n j j2 k ˇ. ; / 2 x Gn x t D n 2 ˇ 4 2 2 n2 ˇ .1 ˇ.k C 1// kŠ 4c t c jxj t kD0 Â Ã n1 p C1 p . 1/ 2 X 1 Á Á jxj C n ˇ : (10.6) .4 2 ˇ/ 2 pC1 ˇ.pCn/ c t D0 1 Š ct2 p 2 n1 2 p 2 For n even and 0<ˇÄ 2 the FS of the time fractional diffusion-wave operator is given by

n 2 Â Ã 1 X2 n k 1 j j2 k ˇ. ; / 2 x Gn x t D n 2 ˇ 4 2 2 n2 ˇ .1 ˇ.k C 1// kŠ 4c t c jxj t kD0 Á 4 2 ˇ Â Ã n C1 C1X n ˇ 1 ˇ n . 1/ c t 2 k .1/ 2 k C 2 k C 2 C k C C ln jxj2 j j x : C n 2 ˇ .4 2 ˇ/ 2 C n 1 ˇ C n Š 4c t c t kD0 k 2 k 2 k (10.7)

10.4 Fractional Moments

For each n 2 the fractional moments Mn .t/ of order >0of the fundamental ˇ solution Gn .x; t/ can be computed explicitly using the Mellin transform (see [FeVi17]) 112 M. Ferreira and N. Vieira

ˇ Table 10.1 Particular moments of Gn n D 1 n D 2 n D 3 n 2 ˇ 2 ˇ 2n n 1 2 1 1 .4c t / 2 2 D ct Á Á 2Á ˇ 2 ˇ ˇ n ˇ.2n/ (mean value) 2 1 4 2 1 2 2 1 C 2 ct 2 C 2 2 ˇ ˇ 2 ˇ 3n 5n 2 2 1 .4c t / 2 D c t ct Á 2 Á .1 ˇ/ ˇ 4 n1 ˇ.3n/ (variance) C 4 1 4 2 1 C 2 C 2 2 ˇ 3 2 ˇ ˇ 2 ˇ 4n 6n 3 3. / 2 2 2 .4c t / 2 D c t Á c t ct Á 2 Á 3ˇ .1 ˇ/ ˇ n ˇ.4n/ (3rd moment) 1 C 1 2 2 1 C 2 C 2 C 2

Á Á Z 2 ˇ nC1 3C n C1 C1 .4c t / 2 2 2 M .t/ D r Gn.r; t/ dr D Á : n n ˇ. C1/ 0 2 2 1 n C 2

In Table 10.1 we present the expression of the moments for some particular values of n and .

10.5 Graphical Representations of the Fundamental Solution

Now we present some plots of the fundamental solution for c D 1; n D 1; 2; and some values of the fractional parameter ˇ: For the one-dimensional case we use ˇ the series representation of G1 since the fundamental solution reduces to a Wright function and we can use the algorithm developed by Luchko in [Lu08].

10.5.1 Case n D 1

For n D 1; the fundamental solution can be written as a Wright function (see [MaPa03]). In fact, putting n D 1 in (10.6) we obtain Â Ã ˇ 1 jxj . ; / ˇ ˇ : G1 x t D ˇ W ;1 ˇ 2ct2 2 2 ct2

ˇ It is known that G1 is a probability density function corresponding to a slow diffusion in the case 0<ˇ< 1and a fast diffusion when 1<ˇ< 2(see [MaEtAl01]). In the slow diffusion case the fundamental solution attains its maximum value at x D 0 (Figure 10.1). 10 Multidimensional Time Fractional Diffusion Equation 113

0.6 t = 0.2 t = 0.2 t = 0.2 t = 0.2 0.5 t = 0.4 0.6 t = 0.4 0.6 t = 0.4 0.6 t = 0.4 t = 0.6 0.5 t = 0.6 0.5 t = 0.6 0.5 t = 0.6 0.4 t = 0.8 t = 0.8 t = 0.8 t = 0.8 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0 0 0 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5-4-3-2-1012345 -5 -4 -3 -2 -1 0 1 2 3 4 5 x x x x 1.2 2 t = 0.2 t = 0.2 t = 0.2 1.8 t = 0.2 0.6 t = 0.4 0.6 t = 0.4 1 t = 0.4 t = 0.4 1.6 t = 0.6 0.5 t = 0.6 0.5 t = 0.6 t = 0.6 t = 0.8 t = 0.8 0.8 t = 0.8 1.4 t = 0.8 0.4 0.4 1.2 0.6 1 0.3 0.3 0.8 0.4 0.2 0.2 0.6 0.4 0.1 0.1 0.2 0.2 0 0 0 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x x x x

ˇ Fig. 10.1 Plots of G1 for c D 1; ˇ D 0:2; 0:5; 0:75; 0:9 (1st line, from left to right), ˇ D 1:0; 1:2; 1:5; 1:7 (2nd line, from left to right), and t D 0:2; 0:4; 0:6; 0:8

10.5.2 Case n D 2

Considering n D 2 in (10.7) the fundamental solution simpliﬁes to

ˇ G2 .x; t/ Á 4c2tˇ Â Ã C1X 2 .k 1/ ˇ .1 ˇ .k 1// 2 2 k 1 C C C ln jxj jxj D : 4 c2 tˇ .k C 1/ .1 ˇ .k C 1// kŠ 4c2tˇ kD0

In this case we use the integral representation (10.3) to make the plots since the series for these cases doesn’t involve exclusively Wright functions. Moreover, the integral representation (10.3) allows us to evaluate the fundamental solution with a better accuracy. In Figure 10.2 we show some plots of the fundamental solution for c D 1 and some ﬁxed values of ˇ and t: From the plots we observe that for ˇ 20; 1Œ the fundamental solution represents a slow diffusion process both in space and time. The decay is more pronounced in space than in time due to the term jxj2 that appears in the series representation of the fundamental solution. For ˇ 21; 2Œ we have a fast diffusion process and its behaviour can be interpreted as propagation of damped waves whose amplitude decreases with time. For ˇ D 1 the plot represents the classical solution of the heat equation. In the limit case ˇ ! 2; it reduces to the fundamental solution of the wave equation. 114 M. Ferreira and N. Vieira

ˇ Fig. 10.2 Plots of G2 for c D 1; ˇ D 0:2; 0:5; 0:75; 0:9 (1st line, from left to right), and ˇ D 1:0; 1:2; 1:5; 1:7 (2nd line, from left to right), with jxj2Œ0:1; 2 and t 2 Œ0:1; 1

10.6 Application to Diffusion in Thermoelasticity

From [Po07] we know that the quasi-static uncoupled theory of diffusive stress results from the equilibrium equation in terms of displacements

v C . C / grad div v D u Ku grad u; the stress-strain-concentration relation

D 2e C . tre u Ku u/I; and the time fractional diffusion equation Á C ˇ 2 @t c u D Q; (10.8) where v is the displacement vector, the stress tensor, e the linear strain tensor, u the concentration, Q the mass source, c2 the diffusivity coefﬁcient, and are Lamé 2 constants, Ku D C 3 , u is the diffusion coefﬁcient of volumetric expansion, and I denotes the unit tensor. Taking into account Parkus (1959) and Nowacki (1986), the author in [Po07] used the representation of the non-zero components of the stress tensor in terms of displacements potential

D 2 .rr I / ; (10.9) where r is the gradient operator, and results from the equation

1 C D mc; m D u ; 1 3 10 Multidimensional Time Fractional Diffusion Equation 115 where is the Poisson ratio. In [Po07], equation (10.8) was subject to the following initial conditions:

@u u.x;0/D P.x/; 0 < ˇ Ä 2; and .x;0/D W.x/; 1 < ˇ Ä 2: (10.10) @t

For the particular case of the two-dimensional axisymmetric case, using (10.8), (10.9) and the initial conditions (10.10) we obtain the following initial-value problem in polar coordinates (see [Po07] for more details): 8 Â 2 Ã ˆ ˇ 2 @ u 1 @u ˆ C@ u D c C C Q.r; t/ < t @r2 r @r u.r;0/D P.r/; 0 < ˇ Ä 2 (10.11) ˆ ˆ @u : .r;0/D W.r/; 1 < ˇ Ä 2 @t and Â Ã @2 1 @ D C D2 ; D 2 : zz rr rr @r2 r @r

In [Po07] it is proved that the solution of (10.11) is given by the formula Z Z Z t C1 C1 u.r; t/ D 2 RQ.R; t/ EQ.r; R; t / dR d C 2 RP.R/ EP.r; R; t/ dR 0 0 0 Z C1 C2 RW.R/ EW .r; R; t/ dR; 0 where EQ.r; R; t/, EP.r; R; t/ and EW .r; R; t/ are three types of fundamental solutions corresponding to q p Q.r; R; t/ D ı.r R/ı .t/; P.r; R/ D ı.r R/; 2 R C 2 R w W.r; R/ D ı.r R/; 2 R with q, p and w being introduced multipliers in order to obtain non-dimensional quantities. A particular case of (10.11)is

8 Â 2 Ã ˆ ˇ 2 @ u 1 @u ˆ C@ u D c C <ˆ t @r2 r @r . ;0/ p ı. /; 0 < ˇ 2 : ˆ u r D r Ä (10.12) ˆ 2 :ˆ @u .r;0/D 0; 1 < ˇ Ä 2 @t 116 M. Ferreira and N. Vieira

For the solution of the Cauchy problem (10.12)wehave(see[Po07]) Z C1 p 2 2 ˇ u.r; t/ D Eˇ.c t / J0.r / d (10.13) 2 0 Z C1 p 2 2 ˇ rr D2 m Eˇ.c t / J1.r / d: 2 r 0

It is immediate that integral representation (10.13) coincides with (10.3)forthe case of n D 2: Therefore the solution of (10.13) has the following series representation: Á 4 2 ˇ Â Ã C1X 2 . 1/ ˇ .1 ˇ . 1// c t 2 k p k C k C C ln r2 r u.r; t/ D : 4 c2 tˇ .k C 1/ .1 ˇ .k C 1// kŠ 4c2tˇ kD0 Applying the same techniques presented in [FeVi17] we obtain the following series representation for rr: Á 4 2 ˇ Â Ã C1X .2 / .1 / ˇ .1 ˇ . 1// c t 2 k mp C k C C k k C C ln r2 r D : rr 4 c2 tˇ .k C 2/ .1 ˇ .k C 1// kŠ 4c2tˇ kD0

Acknowledgements The authors were supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project UID/MAT/ 0416/2013. N. Vieira is Auxiliar Researcher, under the FCT Researcher Program 2014 (Ref: IF/00271/2014).

References

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[LoSh67] Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967) [Lu13a] Luchko, Y.: Multi-dimensional fractional wave equation and some properties of its fundamental solution. Commun. Appl. Ind. Math. 6(1), Article ID 485, 21 p. (2013) [Lu13b] Luchko, Y.: Fractional wave equation and damped waves. J. Math. Phys. 54(3), Article ID 031505, 16 p. (2013) [Lu08] Luchko, Y.: Algorithms for evaluation of the Wright function for the real arguments values. Fract. Calc. Appl. Anal. 11(1), 57–75 (2008) [MaPa03] Mainardi, F., Pagnini, G.: The Wright functions as solutions of the time-fractional diffusion equation. Appl. Math. Comput. 141(1), 51–62 (2003) [MaEtAl01] Mainardi, F., Luchko, Y., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4(2), 153–192 (2001) [MeKl00] Metzler, R., Klafter, J.: Accelarated Brownian motion: a fractional dynamics approach to fast diffusion. Europhys. Lett. 51, 492–498 (2000) [MeEtAl98] Metzler, R., Klafter, J., Sokolov, I.M.: Anomalous transport in external ﬁelds: continuous time random walks and fractional diffusion equation extended. Phys. Rev. E 58, 1621–1633 (1998) [Po07] Povstenko, Y.Z.: Two-dimensional axisymmetric stresses exerted by instantaneous pulses and sources of diffusion in an inﬁnite space in a case of time-fractional diffusion equation. Int. J. Solids Struct. 44(7–8), 2324–2348 (2007) [ScWy89] Schneider, W.R., Wyss, W:: Fractional diffusion and wave equations. J. Math. Phys. 30(1), 134–144 (1989) [Wy86] Wyss, W.: The fractional diffusion equation. J. Math. Phys. 27(11), 2782–2785 (1986) Chapter 11 On Homogenization of Nonlinear Robin Type Boundary Conditions for the n-Laplacian in n-Dimensional Perforated Domains

D. Gómez, E. Pérez, A.V. Podol’skii, and T.A. Shaposhnikova

11.1 Introduction

In this paper, we study the asymptotic behavior of the solution u",as" ! 0,ofa boundary value problem for the p-Laplacian. The partial differential equation for the p-Laplace operator is posed in a domain ˝" which is obtained by removing small regions G" of diameter O.a"/, the cavities/perforations, from a fixed domain n ˝ of . Here, n 3, p D n, a" " where " is a small parameter that measures the size of a periodic grid in n, containing the cavities, for which a certain non- periodic distribution over the whole volume is allowed: see Figure 11.1. Each cavity is homothetic to some domain ranging in a fixed set of isoperimetric domains of the space (cf. (11.1)). On the boundary of the cavities S", we consider a generalized @ ˇ."/. ; / 0 Robin boundary condition n u" C x u" D which involves a continuously differentiable function D .x; u/ defined in ˝ , strictly monotone with respect to u, and the order function ˇ."/. ˇ."/ may converge towards 1 as " ! 0, and it is @ n2. ; / referred to as adsorption parameter.Also n u denotes jruj ru where is the unit outward normal vector to the boundary of ˝" (cf. Remark 2). Homogenization problems in porous media for the p-Laplace operator have been considered recently in the literature of applied mathematics: in this connection, we refer to [GoEtAl17] for a long list of references on the subject as well as on models related to the p-Laplacian, for different values of p; see also [GoEtAl17]for a detailed description of the state of the art on the subject.

D. Gómez • E. Pérez () Universidad de Cantabria, Santander, Spain e-mail: [email protected]; [email protected] A.V. Podol’skii • T.A. Shaposhnikova Moscow State University, Moscow, Russia e-mail: [email protected]; [email protected]

Fig. 11.1 The geometrical conﬁguration of ˝" and the periodicity cell

Further specifying, [GoEtAl16, GoEtAl17, ShEtAl12] and [PoEtAl15]arethe closest papers in the literature, since they consider the p-Laplacian and, on the boundary conditions, they consider the same kind of nonlinear function and adsorption parameter ˇ."/ here considered. This parameter may be very small or very large (cf. Remark 2). In [GoEtAl17] and [ShEtAl12] a complete description of the homogenized problems for the p-Laplace operator in perforated media with p 2 Œ2; n/, for different relations between parameters, has been performed; [ShEtAl12] addresses the homogenization of boundary value problems while [GoEtAl17] considers a variational inequality issue from unilateral constraints for the p-Laplacian. Also, a problem for the case where the perforations are placed along a n 1 dimensional manifold is considered in [GoEtAl16]. The fact that p < n makes an important difference FOR the results, geometrical configurations allowed for the perforations and the techniques in these works. The case p D n has been considered in [PoEtAl15] but dealing with only one of the relations between parameters among the nine that we consider here (cf. (11.12)). The same can be said for the case of restrictions on the flux on the boundary of the cavities in [GoEtAl16] or the case of the Laplacian in dimension n D 2 [PeEtAl14]: only one relation between parameters has been considered. In fact, the results in this paper complete that started in [PoEtAl15] and [PeEtAl14] (see Remark 1). We emphasize that considering p D n allows both a certain non-periodical distribution of the perforations and geometries for perforations different from balls: see [BrEtAl16] and [GoEtAl17] for further comments on geometrical configurations as well as for less restrictive nonlinear functions . For linear problems when p D n D 2 and a" D O."/ see, for instance, [CiEtAl88, OlEtAl95b] and [ChEtAl99]. The aim of the paper is to obtain all the possible homogenized problems for the different relations between the parameters ˇ."/, " and a", and more precisely, to characterize the possible relations giving rise to the appearance of a new term in the homogenized problem, the so-called strange term in the literature of homogenization problems. In this respect, we refer to [MaEtAl74, Sp82] and 11 Homogenization of Robin-Type Boundary Conditions 121

[CiEtAl82] for the first works where these terms arise in linear problems, to [Go95] and [Ka89] for the first works with nonlinear Robin boundary conditions on the boundary of the cavities, the Laplacian being the operator under consideration, and to [Po10] for the first work related to the p-Laplacian operator and generalized Robin boundary conditions. When computing the homogenized problems, we obtain critical sizes of perforations or critical relations between parameters in such a way that, depending on whether these relations are satisfied, the strange term can change its character or it may disappear. We deal with such critical relations when one of the limits of " and ", C and ˛, respectively, (cf. (11.12)), are finite numbers; " and " are defined by (11.11). Figure 11.2 provides a map of all the possible homogenization problems for different relations between the parameters. Due to the logarithmic scale, for this representation, we have chosen the parameters a" and ˇ."/ ranging in a large set of different orders of magnitude which provide all the possible values for C and ˛ in (11.12), including 0 and 1. This fact was noticed in [GoEtAl15] for the Laplace operator in a domain perforated by thin tubes, namely, for a very different homogenization problem from what we consider here. However, it is in good agreement with the different critical sizes found in homogenization problems for the Laplacian in porous media depending on whether n D 2 or n >2.The occurrence of a logarithm scale to compute critical sizes was also detected in [LaEtAl89] in the homogenization of the Dirichlet boundary conditions for the p-Laplace operator: see [GoEtAl17, GoEtAl16, PeEtAl14, PoEtAl15, ShEtAl12]to compare results for the p-Laplacian, with generalized Robin conditions/constraints on the boundary of the perforations, when p Ä n. It should be noted that the most critical situation happens when both constants ˛ and C in (11.12) are positive finite numbers. The homogenized problem is that in point I of the table in Section 11.2 (cf. the bold line in Figure 11.2). The strange term appears in the partial differential equation (cf. (11.13)) and it is a function implicitly defined, from , through a functional equation (11.14). Also, the perimeter of the cavities l arises in the averaged constants, as a parameter, for any shape (11.1). Nowadays this is a well-known fact (cf. [GoEtAl17, GoEtAl16, PoEtAl15]), and obtaining this homogenized problem has been the object of study in [PoEtAl15]. We also emphasize that a very important critical relation happens when C is positive and ˛ D 0. That is, when we deal with big sizes of perforations (comparing with the case of finite ˛) and the critical relation for the adsorption parameter, which amounts to the fact that the total area of the perforations multiplied by the adsorption is positive. The homogenized problem is in point II of the table. The strange term in (11.15) is nonlinear but it contains the function multiplied by an averaged constant where the perimeter also arises for any shape of the cavities. As the Figure 11.2 shows (cf. the plane D m=.n 2/) this relation between parameters gives a new critical size of the perforations, for a fixed adsorption, and it also gives somehow a “critical” adsorption parameter, for a fixed size of the perforations. In this way, we pass from a nonlinear strange term containing to the extreme cases where either we have a large adsorption on big perforations or we 122 D. Gómez et al. have very small perforations with any adsorption. Namely we range between points II, IV, or V in the table. On the other hand, if we restrict ourselves to the faces of the triangular prism in Figure 11.2, we always deal with critical relations, changing the character of the strange term across the intersecting line (the bold line). The homogenized problems are given in points I, II, and III of the table. On the upper plane (case of very large adsorption and ˛>0), asymptotically, the behavior of the solution is as if a Dirichlet condition were prescribed on the boundary of the perforations (cf. [LaEtAl89] to compare). Among the techniques used for the approaches in this paper we mention: the theory of monotonic operators, extension operators, the energy method, the construction of the auxiliary functions W" and M" (cf. (11.20) and (11.29)), and results on convergence of measures and comparison of measures (cf. Lemmas 3 and 4). Note that W" are usual functions arising in the homogenization of perforated domains for the p-Laplacian; they can be explicitly computed and allow us to construct the test functions to pass to the limit in weak formulations (cf. Theorems 2–3). In contrast, the proof of the above mentioned comparison results relays in other different auxiliary functions (cf. [GoEtAl16]), which as a matter of fact are used when the perforations are not balls and which cannot be explicitly constructed. The structure of the paper is as follows: the setting of the homogenization problem and the table of homogenized problems are in Section 11.2. Section 11.3 gathers all the preliminary results that we use to derive the convergence of the extension of the solutions in the space W1;n. Sections 11.4 and 11.5 contain the case in which we deal with either a critical relation for the adsorption parameter or a critical size of the perforations. A strange term arises in the partial differential equations and the convergence (cf. Theorems 1 and 2, respectively) holds in the weak topology of W1;n. Section 11.6 contains the rest of the cases, the solutions ignore asymptotically either adsorption or perforations, and as a consequence we may have stronger results of convergence or even bounds for convergence rates (cf. Theorems 3–5). We refer to [GoEtAl17] for the technique to obtain correctors and strong convergence.

11.2 Setting of the Problem and Homogenized Problems

Let ˝ be a bounded domain in n, n 3, with a smooth boundary @˝.Let" be a small positive parameter that we shall make converge towards zero. We set e ˝" Dfx 2 ˝ W .x;@˝/>2"g where denotes the distance. Let M be a ﬁnite subset of which we can identify with f1; 2; ; mMg for m mM 2 . Assume that we have the set M of domains D satisfying the following n properties: for any m 2 M, Dm T1=4 Y, where Y D .1=2; 1=2/ , T1=4 Dfy 2 n Wjyj < 1=4g, Dm is diffeomorphic to a ball m 2 M,andtheareaofDm is equal to a given number l >0, i.e. 11 Homogenization of Robin-Type Boundary Conditions 123

j@DmjDl; 8m 2 M: (11.1)

Let a" ", we deﬁne [ [ j j G" D .a"G C "j/ D G"; j2" j2"

j m n j where G coincides with one of the domains D , m 2 M, and " Dfj 2 W G" j j e n Y" D "Y C "j; G" \ ˝" ¤;g(see Figure 11.1). Obviously, we have j"jŠd" , with some d >0, and

j j j j ; G" Ta" T"=4 Y"

j j "=4 j where Ta" and T"=4 are balls with radius a" and , respectively, and center P", j which coincides with the center of Y". Now we can deﬁne

˝" D ˝ n G"; S" D @G";@˝" D @˝ [ S":

1;n 1;n Also, we consider the space W .˝";@˝/ (W .˝; @˝/, respect.) to be the 1;n 1;n completion with respect to W .˝"/-norm (W .˝/-norm, respect.) of the set of inﬁnitely differentiable functions in ˝" (˝, respect.), vanishing in a neighborhood of @˝. Let us consider .x; u/ a continuously differentiable function of variables .x; u/ 2 ˝ satisfying:

.x;0/D 0; (11.2) n ..x; u/ .x; v//.u v/ k1ju vj ; (11.3)

n1 j.x; u/jÄk2juj ; (11.4) for all x 2 ˝; u;v 2 , and certain constants k1 >0, k2 >0. Note that (11.2)– (11.4)imply

.x; u/ 0 if u 0 and .x; u/ Ä 0 if u Ä 0; 8x 2 ˝:

For f 2 Lq.˝/ with q D n=.n 1/, we consider the following problem : 8 < nu" D f in ˝"; @ u" ˇ."/.x; u"/ 0 x S"; (11.5) : n C D for 2 u" D 0 on @˝;

v. n2 / @ n2. ; / where nu Á di jruj ru , n u Ájruj ru , denotes the unit outward normal to ˝" on S" and "-dependent constant ˇ."/ > 0. 124 D. Gómez et al.

1;n The variational formulation of problem (11.5) is: ﬁnd u" 2 W .˝";@˝/ satisfying Z Z Z n2 1;n jru"j ru"rvdx C ˇ."/ .x; u"/vds D f vdx; 8v 2 W .˝";@˝/:

˝" S" ˝" (11.6) The existence and uniqueness of the solution of problem (11.6) follows from the monotonicity of the functions jujn2u and .x; u/ with respect to u (see, e.g., Section II.8.2 in [Li69]). Moreover, applying Minty Lemma, (11.6) amounts to the inequality Z Z Z n2 jrvj rvr.v u"/dx C ˇ."/ .x; v/.v u"/ds f .v u"/dx;

˝" S" ˝" (11.7)

1;n where v is an arbitrary function from W .˝";@˝/. 1;p Let us denote byeu" an extension of u" to ˝,eu" 2 W .˝; @˝/, satisfying:

e 1; 1; ; e : ku"kW n.˝/ Ä Cku"kW n.˝"/ kru"kLn.˝/ Ä Ckru"kLn.˝"/ (11.8) We refer to Theorem 2 in [Po15] and Theorem 1 in [PoEtAl15] for the construction of such an extension. Taking into account (11.8) and the properties of the function .x; u/,wesetin(11.6) v Á u" and we obtain the estimations for the solution u":

n n q e" 1; ˇ."/ " : ku kW n.˝/ C ku kLn.S"/ Ä Kkf kLq.˝"/ (11.9)

Thus, there is a subsequence (still denoted by ") such that, as " ! 0,

1;n n eu" * u in W .˝; @˝/ weak and eu" ! u in L .˝/; (11.10) for a certain function u which, once identiﬁed, provides the convergences (11.10) for the whole sequence of ". In what follows, we show that this function u is the unique solution of a homogenized problem which depends on the relation between the size of the holes a", the period ", and the adsorption parameter ˇ."/: As a matter of fact, if we denote by " and " the following relations

n n1 n 1 " D ˇ."/a" " and " D " n j ln.a"/j; (11.11) and consider the nine possibilities for the couple of limits

lim " D C and lim " D ˛; (11.12) "!0 "!0 where C and ˛ are well-determined positive constants, 0 or 1, we obtain ﬁve possible different limit behaviors of eu". 11 Homogenization of Robin-Type Boundary Conditions 125

Only one of these cases has been considered previously in the literature (cf. Remark 1). This case, when both limits C and ˛ are strictly positive ﬁnite numbers, can be referred to as the most critical case (see [PoEtAl15] for the proof). Here, we consider the rest of the cases. In order to make more comprehensible the entire results, below we introduce a table with all the possible limit situations and a map (cf. Figure 11.2) with a sketch of all these possibilities. The existence and uniqueness of the solution for all the homogenized problems holds as does that for the "-dependent problem (11.5).

11.2.1 Table of Homogenized Problems

I. When " ! C >0and " ! ˛>0, the homogenized problem is: u C A jH.x; u/jn2H.x; u/ D f in ˝; n n (11.13) u D 0 on @˝;

1n where An D !n˛ and, for every .x;/ 2 ˝ , H.x;/is the solution of the functional equation

n2 BnjHj H D .x; H/; (11.14)

1n 1 1 n with Bn D !n˛ C l , !n denotes the area of the unit sphere in , and j@GjjDl. II. When " ! C >0and " ! 0, the homogenized problem is u C D.x; u/ D f in ˝; n (11.15) u D 0 on @˝;

where D D Cl: III. When " !1and " ! ˛>0, the homogenized problem is u C A jujn2u D f in ˝; n n (11.16) u D 0 on @˝;

1n n where An D !n˛ , !n denotes the area of the unit sphere in . IV. When " !1or " ! 0, the homogenized problem is u D f in ˝; n (11.17) u D 0 on @˝:

V. When " !1and " ! 0; u Á 0, that is, as " ! 0, the solution u" vanishes asymptotically in the whole ˝. 126 D. Gómez et al.

Fig. 11.2 Sketch of homogenized problems depending on ; ; m in (11.18): see Points I–V in the table

As for the above mentioned map, it should be noted that even in the case of ﬁnite strictly positive limits C and ˛,(11.12) allows many different orders of magnitude for the size a" and the parameter ˇ."/ to be considered. To illustrate the limit behaviors, we choose a large set of different orders of magnitude for these parameters, but ranging in the functions

˛=" nm ˛.n1/=" a" D " e and ˇ."/ D " e ; (11.18) for any constants >0, 0, and m 0. In Figure 11.2, we choose , and m as the coordinate axis. Figure 11.2 shows a general situation for the different relations.

11.3 Preliminary Results

In this section, we introduce certain functions which allow us to construct the test functions to pass to the limit in (11.7), as " ! 0. Also, we obtain certain estimates that we need for proofs in Sections 11.4–11.6. 1;n j Let us introduce the function W" of W .˝/ as follows: Let P" be the center of j " " j j the cube Y" D Y C j and we denote by Tr the ball of radius r with center P".For j 2 ", let us consider the auxiliary problem 8 ˆ j j j < nw" D 0 in T"=4 n Ta" ; j j " 1 @ ; (11.19) ˆ w D on Ta" : j j w" D 0 on @T"=4; 11 Homogenization of Robin-Type Boundary Conditions 127 and we introduce the function deﬁned in ˝ by 8 ˆ j j j < w".x/ if x 2 T"=4 n Ta" ; j 2 "; W".x/ D 1 j ; ; (11.20) ˆ if x 2 Ta" Sj 2 " : 0 ˝ j : if x 2 n j2" T"=4

The solution of (11.19) can be constructed explicitly:

.4 j ="/ j ln jx P"j w".x/ D : (11.21) ln.4a"="/

Thus, we can compute

n "n . ="/ 1n 1n krW"kLn.˝/ Ä K j ln a" j Ä Kj "j (11.22) and, consequently, as " ! 0,

1;n W" *0in W .˝/ weak if " ! ˛>0; 1;n (11.23) W" ! 0 in W .˝/ if " !1:

Below we introduce here certain results which prove to be useful for the proofs throughout Sections 11.4–11.6. e e n n Lemma 1 Let Y" be Y" D ".1=2; 1=2/ n a"G0 where G0 is a domain of n diffeomorphic to a ball, n 3, and 0

1;n Lemma 2 Let h 2 W .˝";@˝/,n 2. Then Z Z ˇ ˇ Z n ˇ " ˇn1 o n 1n n n ˇ ˇ n n jhj dx Ä K a" " jhj ds C ˇln ˇ " jrhj dx : 2a" ˝" S" ˝"

1 1 j Lemma 3 Let h" 2 H0 .˝/ and h" * h0 in H .˝/-weak as " ! 0. Let T"=4 be the j ball of radius "=4 with center P": Then, as " ! 0, X Z Z 2.n1/ 2 " h" ds ! !n h0 dx; j2 " j ˝ @T"=4

n where !n is the surface area of the unit sphere in . 128 D. Gómez et al.

1;n.˝; @˝/ Lemma 4 Let us assume h" 2 W where kh"kW1;n.˝/ is bounded indepen- " j j : dently of . Let Ta" be the ball of radius a" with center P" Then, ˇ ˇ ˇ ˇ ˇZ X Z ˇ ˇ l ˇ 1 n1 ˇ ˇ . " / 1; ; ˇ h"ds h"dsˇ Ä K a" kh"kW n.˝/ !n ˇ j2" ˇ S" j @Ta"

n j where !n is the surface area of the unit sphere in and l Dj@G j for any j 2 ". Lemma 5 Let p be p >2and n 3. Let v 2 W1;1.˝/, ' 2 W1;p.˝; @˝/ and 1;p " 2 W .˝; @˝/ such that kr"kLm.˝/ ! 0,as" ! 0,form2 Œ1; p/. Then, Z p2 jr.v "/j r.v "/r'dx

˝" Z Z (11.24) p2 p2 D jrvj rvr'dx jr"j r"r'dx C R";

˝" ˝" where jR"j!0 as " ! 0. Moreover, if kr"kLp.˝/ ! 0,as" ! 0, then Z Á p2 p2 lim jr.v "/j r.v "/ jrvj rv r'dx D 0: (11.25) "!0 ˝"

In addition, (11.24) and (11.25) also hold in the case where ' depends on ", namely ' Á '", with kr'"kLp.˝/ bounded independently of ". In the above lemmas, and in what follows, K denotes a constant independent of ". Also, in these lemmas, the constant K does not depend on the functions w; h appearing in their statements. The proof of Lemmas 1 and 2 holds applying the techniques in Lemma 2 in [OlEtAl95a] and Lemma 3 in [OlEtAl96], respectively, for p D 2. We refer to Lemma1in[ZuEtAl11] and Lemma 3 in [GoEtAl16] for the proof of Lemmas 3 and 4, respectively. See Proposition 2.3 in [GoEtAl17] for the proof of Lemma 5 (cf. also [Po10]).

11.4 Critical Relation for the Adsorption

In this section, we consider the case C >0and ˛ D 0 in (11.12) to obtain the homogenized problem (cf. point II in the table); the strange term arises in the partial differential equation, and it is a nonlinear function involving , namely the function 11 Homogenization of Robin-Type Boundary Conditions 129 appearing in the generalized Robin condition on the perforations, and averaged constants of the problem which include the perimeter of the cavities l.

Theorem 1 Let " ! C >0and " ! 0 when " ! 0, and let u" be the weak solution of (11.5). Then, the limit function u of the extension of u", deﬁned by (11.10), is the weak solution of problem (11.15). 1 Proof Let us pass to the limit in (11.7)as" ! 0 with v 2 C0 .˝/. Since eu" * u 1;n n n in W .˝; @˝/ and jG"jDO.a"" / with a" ", we get Z Z n2 n2 lim jrvj rvr.v u"/dx D jrvj rvr.v u/dx (11.26) "!0 ˝" ˝ and Z Z

lim f .v u"/dx D f .v u/dx: (11.27) "!0 ˝" ˝

Let us prove that under the conditions stated in the theorem we have Z Z

lim ˇ."/ .x; v/.v u"/ds D D .x; v/.v u/dx; (11.28) "!0 S" ˝ where D D Cl: We follow the method that was introduced in [ShEtAl12]. j j For any j 2 M, we deﬁne the function M".x/ on the cell Y" as a solution of the problem 8 ˆ j ej j j < nM" D " in Y" D Y" n G"; j j j @ M" D 1 on @G" D S" (11.29) :ˆ n @ j 0 @ej @ j ; n M" D on Y" n G"

n1 n j a" " j@G j j j where " D 1 . Let us assume hM"i j D 0: For the function M".x/ we 1.a"" /njGjj eY" have

1 j krM k j 6 Ka"j ln."=a"/j n : (11.30) " Ln.eY"/

b" ej We denote by Y D[j2" Y" and introduce the function M" such that M".x/ D j ej M".x/ if x 2 Y"; j 2 ". Then, from (11.30) we have an estimation of the gradient b" of the function M".x/ on Y :

1 1 " 6 "" ."= "/ n : krM kLn.bY"/ Ka j ln a j (11.31) 130 D. Gómez et al.

By means of M", the integral on S" can be transformed into a volume integral. Thus, we can write Z X Z j n2 j ˇ."/ .x; v/.v u"/ds Dˇ."/ div.jrM"j rM".x; v/.v u"//dx 2" " j j S eY" X Z j n2 j Dˇ."/ jrM"j rM"r..x; v/.v u"//dx j2 " j eY" X Z C ˇ."/ " .x; v/.v u"/dx: j2 " j eY" (11.32) Now, from (11.9), (11.31) and the assumptions " ! C >0and " ! 0 as " ! 0; we can estimate the ﬁrst term in the right-hand side of (11.32), that is, Z ˇ."/ n1 .. ; v/.v // 6 ˇ."/ n1 v e jrM"j jr x u" jdx K krM"k k u"kW1;n.˝/ Ln.bY"/

bY"

n1 .n1/ .n1/=n .n1/=n 6Kˇ."/a" " j ln."=a"/j 6 K"" ! 0; as " ! 0: (11.33) b" In addition, using the deﬁnition of ", the measure of ˝ n Y and (11.10), we have X Z Z j lim ˇ."/ " .x; v/.v u"/dx D Cj@G j .x; v/.v u/dx: (11.34) "!0 j2 " j ˝ eY"

Then, gathering (11.32), (11.33), and (11.34) yields (11.28). 1 Thus, passing to the limit in (11.7) with v 2 C0 .˝/ and using (11.26), (11.27), and (11.28), we obtain the following variational inequality Z Z Z jrvjn2rvr.v u/dx C D .x; v/.v u/dx > f .v u/dx; (11.35) ˝ ˝ ˝

1 1;n for all v 2 C0 .˝/ and, by density, for all v 2 W .˝; @˝/: Setting v D u ˙ in (11.35) with >0and an arbitrary function from W1;n.˝/, and taking limits as !C0, we obtain that u is the solution of Z Z Z jrujn2rur dx C D .x; u/ dx D f dx; 8 2 W1;n.˝; @˝/: ˝ ˝ ˝ 11 Homogenization of Robin-Type Boundary Conditions 131

Therefore, the limit (11.10) holds for all the sequence of ", and this limit u is the weak solution of problem (11.15). This concludes the proof of the theorem.

11.5 Critical Size for Perforations

In this section, we consider the case C D1and ˛>0in (11.12) to obtain the homogenized problem. The strange term arises in the partial differential equation, being a classical reaction term associated with the n-Laplacian, namely of the type jujn2u, as if Dirichlet condition were imposed on the boundary of the perforations instead of the generalized Robin one.

Theorem 2 Let " !1and " ! ˛>0when " ! 0, and let u" be the weak solution of (11.5). Then, the limit function u of the extension of u", deﬁned by (11.10), is the weak solution of problem (11.16). 1 Proof Let us take v D '.1W"/ in the integral inequality (11.7) where ' 2 C0 .˝/ and W" is deﬁned by (11.20). Since W" D 1 in G" and .x;0/D 0, we obtain Z Z n2 jr.' W"'/j r.' W"'/r.' W"' u"/dx f .' W"' u"/dx:

˝" ˝" (11.36) On account of (11.23) and (11.10), we deduce Z Z

lim f .' W"' u"/dx D f .' u/dx: (11.37) "!0 ˝" ˝

Let us show Z n2 lim jr.' W"'/j r.' W"'/r.' W"' u"/dx "!0 ˝" Z Z (11.38) n2 n2 D jr'j r'r.' u/dx C An j'j '.' u/dx; ˝ ˝

n1 where An D !n=˛ . From (11.21) we deduce

m "m . ="/ m "m=.n1/ m Œ2; /: krW"kLm.˝/ Ä K j ln a" j D K j "j for m 2 n (11.39) 132 D. Gómez et al.

Then, using (11.39), (11.22), and (11.9), we apply Lemma 5 and obtain that Z n2 lim jr.' W"'/j r.' W"'/r.' W"' u"/dx "!0 ˝" Z n2 D lim jr'j r'r.' W"' u"/dx "!0 (11.40) ˝" Z n2 lim jr.W"'/j r.W"'/r.' W"' u"/dx: "!0 ˝"

Taking into account (11.10), (11.23), (11.39) and the size of G", we can prove that Z Z n2 n2 lim jr'j r'r.' W"' u"/dx D jr'j r'r.' u/dx (11.41) "!0 ˝" ˝ and Z n2 lim jr.W"'/j r.W"'/r.' W"' u"/dx "!0 ˝" Z Á (11.42) n2 n2 D lim jrW"j rW"r j'j '.' W"' u"/ dx: "!0 ˝"

Besides, by deﬁnition of W" and the Green formula, we have that Z Á n2 n2 jrW"j rW"r j'j '.' W"' u"/ dx

˝" X Z j n2 j n2 D jrw"j @ w"j'j '.' u"/ds (11.43) j2 " j @T"=4 X Z j n2 j n2 jrw"j @ w"j'j 'u"ds; j2 " j @Ta" where @ g denotes the normal derivative of g. Now, we study the limit of the last two integrals taking into account that, by (11.21), ˇ 4 ˇ 1 j ˇ j ˇ @ w"ˇ D and @ w"ˇ D : (11.44) @ j " . 4a" / @ j . 4a" / T"=4 ln " Ta" a" ln " 11 Homogenization of Robin-Type Boundary Conditions 133

Firstly, we write ˇ ˇ ˇ ˇ ˇ Z ˇ ˇX ˇ ˇ j n2@ j ' n2' ˇ ˇ jrw"j w"j j u"dsˇ ˇj2 " j ˇ @Ta" ˇ ˇ ˇ Z Z ˇ ! ˇ X ˇ n ˇ l n2 n2 ˇ 6 ˇ j'j 'eu"ds j'j 'eu"dsˇ n1 n1 ˇ ˇ la" j ln.4a"="/j !n ˇ j2" ˇ j S" @Ta" ˇ ˇ ˇZ ˇ ! ˇ ˇ n ˇ n2 ˇ ' 'eu"ds : C n1 n1 ˇ j j ˇ la" j ln.4a"="/j ˇ ˇ S"

Now, from Lemma 4,(11.9) and the size of S", we conclude ˇ ˇ ˇ Z Z ˇ ! ˇ X ˇ " n ˇ l n2 n2 ˇ K ˇ j'j 'eu"ds j'j 'eu"dsˇ Ä n1 n1 ˇ ˇ n1 la" j ln.4a"="/j !n " ˇ j2" ˇ j S" @Ta" and ˇ ˇ ˇZ ˇ ˇ ˇ .n1/=n ! ! jS"j keu"k n. / K n ˇ ' n2'e ˇ n L S" : 1 1 ˇ j j u"dsˇ Ä 1 1 Ä 1= n .4 ="/ n n .4 ="/ n n1 n la" j ln a" j ˇ ˇ la" j ln a" j " " S"

Consequently, ˇ ˇ ˇ ˇ ˇX Z ˇ ˇ j n2 j n2 ˇ lim ˇ jrw"j @ w"j'j 'u"dsˇ D 0: (11.45) "!0 ˇ ˇ ˇj2 " j ˇ @Ta"

Secondly, from (11.44), Lemma 3 and (11.10), we have X Z Z j n2 j n2 n2 lim jrw"j @ w"j'j '.' u"/ds DAn j'j '.' u/dx: "!0 j2 " j ˝ @T"=4 (11.46) Thus, gathering (11.40), (11.41), (11.42), (11.43), (11.45), and (11.46) yields (11.38). Finally, we use (11.37) and (11.38) to pass to the limit in (11.36), as " ! 0, and obtain that the limit function u satisﬁes the following inequality: 134 D. Gómez et al. Z Z Z n2 n2 jr'j r'r.' u/dx C An j'j '.' u/dx f .' u/dx; (11.47) ˝ ˝ ˝

1 1;n for all ' 2 C0 .˝/, and by density, for all ' 2 W .˝; @˝/. Now, taking ' D u˙v in (11.47) where v 2 W1;n.˝; @˝/ and passing to the limit as !C0, we obtain that u satisﬁes the integral identity Z Z Z n2 n2 1;n jruj rurvdx C An juj uvdx D f vdx; 8v 2 W .˝; @˝/; ˝ ˝ ˝ which concludes the proof.

11.6 Extreme Cases

In this section, we consider the cases ˛ D1or C D 0 or ˛ D 0 and C D1 in (11.12). To obtain the homogenized problem, either the solution asymptotically ignores adsorption and cavities due to the fact that sizes of cavities are very small or the adsorption parameter is very small, or it vanishes asymptotically.

Theorem 3 Let " !1when " ! 0, and let u" be the weak solution of (11.5). Then, the limit function u of the extension of u", deﬁned by (11.10), is the weak solution of problem (11.17). 1 Proof Let us take v D '.1W"/ in the integral inequality (11.7) where ' 2 C0 .˝/ and W" is deﬁned by (11.20). Since W" D 1 in G" and .x;0/D 0, we obtain Z Z n2 jr.' W"'/j r.' W"'/r.' W"' u"/dx f .' W"' u"/dx

˝" ˝" and we pass to the limit when " ! 0: On account of (11.23) and (11.10), we deduce Z Z

lim f .' W"' u"/dx D f .' u/dx: "!0 ˝" ˝

Besides, using (11.39), (11.23), (11.9), and (11.10), we apply Lemma 5 and we have Z n2 lim jr.' W"'/j r.' W"'/r.' W"' u"/dx "!0 ˝" Z D jr'jn2r'r.' u/dx: ˝ 11 Homogenization of Robin-Type Boundary Conditions 135

Thus, we get that u satisﬁes the following inequality Z Z jr'jn2r'r.' u/dx f .' u/dx; ˝ ˝

1 1;n for all ' 2 C0 .˝/, and by density, for all ' 2 W .˝; @˝/. Now, taking ' D u ˙ v where v 2 W1;n.˝; @˝/ and passing to the limit as !C0,we obtain that u satisﬁes the integral identity Z Z jrujn2rurvdx D f vdx; 8v 2 W1;n.˝; @˝/; ˝ ˝ which concludes the proof.

Theorem 4 Let " ! 0 when " ! 0, and let u" be the weak solution of (11.5). Then, the limit function u of the extension of u", deﬁned by (11.10), is the weak solution of problem (11.17). 1 Proof We take v D ' 2 C0 .˝/ in the integral inequality (11.7) and pass to the limit as " ! 0. On account of (11.10) and the fact that jG"j!0 as " ! 0,we conclude Z Z n2 n2 lim jr'j r'r.' u"/dx D jr'j r'r.' u/dx (11.48) "!0 ˝" ˝ and Z Z

lim f .' u"/dx D f .' u/dx: (11.49) "!0 ˝" ˝

n1 n Moreover, using (11.4), (11.9), jS"jÄKa" " and the fact that " ! 0 as " ! 0, we get ˇ ˇ ˇ Z ˇ ˇ ˇ n1 ˇˇ."/ . ; '/.' / ˇ 6 ˇ."/. n / ˇ x u" dsˇ K jS"jCjS"j ku"kLn.S"/ ˇ ˇ S"

1 n1 n n1 n n Ä K.ˇ."/a" " C .ˇ."/a" " / n / ! 0; (11.50) as " ! 0. Hence, from (11.48), (11.49), and (11.50), we pass to the limit, as " ! 0, 1 in (11.7) with v D ' 2 C0 .˝/ and have that the limit function u satisﬁes the following integral inequality 136 D. Gómez et al. Z Z jr'jn2r'r.' u/dx f .' u/dx; ˝ ˝

1 1;n for all ' 2 C0 .˝/, and by density, for all ' 2 W .˝; @˝/. Taking ' D u ˙ v in (11.47) where v 2 W1;n.˝; @˝/ and passing to the limit as !C0, we obtain that u satisﬁes the integral identity Z Z jrujn2rurvdx D f vdx; 8v 2 W1;n.˝; @˝/; ˝ ˝ which concludes the proof.

Theorem 5 Let " !1and " ! 0 when " ! 0. Then, the extension eu" of the weak solution of (11.5), deﬁned by (11.10), veriﬁes

1 1 1 e 6 . n / n2 ; ku"kW1;n.˝/ K " C "

1;n and consequently,eu" converges to zero in W .˝/ as " ! 0. n " Proof Let us estimate ku kLn.˝"/. Using Lemma 2 and (11.9) we obtain

n 6 . 1n"n n ."=2 / n1"n n / ku"kLn.˝"/ K a" ku"kLn.S"/ Cjln a" j kru"kLn.˝"/ 1n n 1 n1 n 1 n1 6 K.a" " ˇ ."/ Cjln."=2a"/j " / 6 K." C " / and, consequently,

n 0: lim ku"kLn.˝"/ D "!0

1;n Besides, setting v D u" 2 W .˝";@˝/in integral identity (11.6) and using (11.3) we get

n n ˇ."/ =. 1/ n : kru"kLn.˝"/ C ku"kLn.S"/ Äkf kLn n .˝"/ku"kL .˝"/

Thus,

1 1 1 n 6 . n / n ; kru"kLn.˝"/ K " C " and, by (11.8), the theorem holds. Remark 1 Note that all the results of the paper apply to the case in which n D p D 2, with the suitable modiﬁcations; this is due mainly to the nonlinear exponent n 2 which reads 0 in the case of the Laplace operator. Consequently, 11 Homogenization of Robin-Type Boundary Conditions 137 the results of this paper extend those in [PeEtAl14] and [PoEtAl15] where the problem (11.5) has been considered when n D p D 2 and n D p >2respectively, but only in the most critical case where C >0and ˛>0(cf. (11.12) and item I of the table). Remark 2 Note that in the case of the p-Laplace operator, with any p, in problem (11.5), we call the boundary condition on S" generalized Robin condition to differentiate it from the “classical” Robin boundary condition for the p-Laplacian where .x; u/ reads .x; u/ D a.x/jujp1u. Obviously, all the computations apply to this case of classical Robin boundary condition on the boundary of the perforations, containing a large parameter, and, to our knowledge, the homogenization of this problem has not been considered in the literature. In this respect, we also outline that the case where ˇ."/ is a constant has not been addressed previously in the homogenization of porous media with “tiny perforations,” and it is included in our study. Finally, we note that the hypothesis on (cf. (11.3)–(11.4)) can be weakened: see [GoEtAl17]forthep-Laplace operator when p < n and [BrEtAl16]forthe Laplace operator.

Acknowledgements This work has been partially supported by MINECO:MTM2013-44883-P.

References

[BrEtAl16] Brillard, A., Gómez, D., Lobo, M., Pérez, E., Shaposhnikova, T.A.: Boundary homogenization in perforated domains for adsorption problems with an advection term. Appl. Anal. 95, 1517–1533 (2016) [ChEtAl99] Chechkin, G.A., Piatnitski, A.L.: Homogenization of boundary value problem in a locally periodic perforated domain. Appl. Anal. 71, 215–235 (1999) [CiEtAl88] Cioranescu, D., Donato, P.: Homogénéisation du problème de Neumann non homogène dans des ouverts perforés. Asymptot. Anal. 1, 115–138 (1988) [CiEtAl82] Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs I & II. In: Brezis, H., Lions, J.L. (eds.) Nonlinear Partial Differential Equations and Their Applications, Collège de France Séminar, Volume II & III. Research Notes in Mathematics, vols. 60, 70, pp. 98–138, 154–178. Pitman, London (1982) [GoEtAl15] Gómez, D., Lobo, M., Pérez, E., Shaposhnikova, T.A., Zubova, M.N.: On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems. Math. Methods Appl. Sci. 38, 2606–2629 (2015) [GoEtAl16] Gómez, D., Lobo, M., Pérez, M.E., Podolskii, A.V., Shaposhnikova, T.A.: Homog- enization of a variational inequality for the p-Laplacian in perforated media with nonlinear restrictions for the flux on the boundary of isoperimetric perforations: p equal to the dimension of the space. Dokl. Math. 93, 140–144 (2016) [GoEtAl17] Gómez, D., Lobo, M., Pérez, E., Podolskii, A.V., Shaposhnikova, T.A.: Average of unilateral problems for the p-Laplace operator in perforated media involving large parameters on the restriction, ESAIM: Control Optim. Calc. Var. doi: 10. 1051/cocv/2017026 [Go95] Goncharenko, M.: The asymptotic behaviour of the third boundary-value problem solutions in domains with fine-grained boundaries. In: Homogenization and Applications to Material Sciences. GAKUTO International Series. Mathematical Sciences and Applications, vol. 9, pp. 203–213. Gakkotosho, Tokyo (1995) 138 D. Gómez et al.

[Ka89] Kaizu, S.: The Poisson equation with semilinear boundary conditions in domains with many tiny holes. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36, 43–86 (1989) [LaEtAl89] Labani, N., Picard, C.: Homogenization of a nonlinear Dirichlet problem in a periodically perforated domain. In: Recent Advances in Nonlinear Elliptic and Parabolic Problems. Pitman Research Notes in Mathematics Series, vol. 208, pp. 294–305. Longman Sci. Tech., Harlow (1989) [Li69] Lions, J.L.: Quelques méthodes de résolution des Problèmes aux Limites non Linéaires. Dunod, Paris (1969) [MaEtAl74] Marchenko, V.A., Khruslov, E.Ya.: Boundary Value Problems in Domains with a Fine-grained Boundary. Izdat. Naukova Dumka, Kiev (1974) [in Russian] [OlEtAl95a] Oleinik, O.A., Shaposhnikova, T.A.: On homogenization problem for the Laplace operator in partially perforated domains with Neumann’s condition on the boundary of cavities. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 6, 133–142 (1995) [OlEtAl95b] Oleinik, O.A., Shaposhnikova, T.A.: On the averaging problem for a partially perforated domain with a mixed boundary condition with a small parameter on the cavity boundary. Differ. Equ. 31, 1086–1098 (1995) [OlEtAl96] Oleinik, O.A., Shaposhnikova, T.A.: On homogenization of the Poisson equation in partially perforated domains with arbitrary density of cavities and mixed type conditions on their boundary. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. (9) 7, 129–146 (1996) [PeEtAl14] Pérez, M.E., Shaposhnikova, T.A., Zubova, M.N.: Homogenization problem in a domain perforated by tiny isoperimetric holes with nonlinear Robin type boundary conditions. Dokl. Math. 90, 489–494 (2014) [Po10] Podolskii, A.V.: Homogenization limit for the boundary value problem with the p-Laplace operator and a nonlinear third boundary condition on the boundary of the holes in a perforated domain. Dokl. Math. 82, 942–945 (2010) [Po15] Podolskii, A.V.: Solution continuation and homogenization of a boundary value problem for the p-Laplacian in a perforated domain with a nonlinear third boundary condition on the boundary of holes. Dokl. Math. 91, 30–34 (2015) [PoEtAl15] Podolskii, A.V., Shaposhnikova, T.A.: Homogenization for the p-Laplacian in a n-dimensional domain perforated by ﬁne cavities, with nonlinear restrictions on their booundary, when p D n. Dokl. Math. 92, 464–470 (2015) [Sp82] Sanchez-Palencia, E.: Boundary value problems in domains containing perforated walls. In: Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. III. Research Notes in Mathematics, vol. 70, pp. 309–325. Pitman, Boston (1982) [ShEtAl12] Shaposhnikova, T.A., Podolskii, A.V.: Homogenization limit for the boundary value problem with the p-Laplace operator and a nonlinear third boundary condition on the boundary of the holes in a perforated domain. Funct. Differ. Equ. 19, 351–370 (2012) [ZuEtAl11] Zubova, M.N., Shaposhnikova, T.A.: Homogenization of boundary value problems in perforated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem. Differ. Equ. 47, 78–90 (2011) Chapter 12 Interior Transmission Eigenvalues for Anisotropic Media

A. Kleefeld and D. Colton

12.1 Introduction

In recent years transmission eigenvalues have come to play an important role in inverse scattering theory and for a survey of results in this area we refer the reader to the recently published monograph [CaCoHa17]. The interest in transmission eigenvalues lies in the fact that they can be determined from the measured scattering data and carry information about the constitutive parameters of the scattering object. In particular, changes in the constitutive parameters of the media result in corresponding changes in the measured transmission eigenvalues and hence transmission eigenvalues can play an important role in the nondestructive testing of inhomogeneous media. This fact is particularly relevant in the case of anisotropic media for which very few methods exist for the nondestructive testing of materials. The use of transmission eigenvalues in determining estimates for the constitutive parameters of an anisotropic media is based on the monotonicity properties of transmission eigenvalues [CaCoHa17]. In particular, upper and lower bounds for inhomogeneous constitutive parameters can be found by determining constant values of the constitutive parameters that yield approximately the same transmission eigenvalue as the one measured for the inhomogeneous case. For details of how this procedure works we refer the reader again to the monograph [CaCoHa17]. Of interest to us in this paper is that the entire procedure for estimating the constitutive parameters of an inhomogeneous anisotropic media from measured transmission eigenvalues rests on being able to determine transmission eigenvalues

A. Kleefeld () Supercomputing Centre, Forschungszentrum Jülich GmbH, Jülich, Germany e-mail: [email protected] D. Colton Department of Mathematical Sciences, University of Delaware, Newark, DE, USA e-mail: [email protected]

© Springer International Publishing AG 2017 139 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_12 140 A. Kleefeld and D. Colton for homogeneous anisotropic materials. To our knowledge the only previous papers on the numerical analysis for the computation of transmission eigenvalues for homogeneous anisotropic media are those by Cakoni, Colton, Monk and Sun [CaEtAl10] and Monk and Sun [MoSu12], both of which use finite element methods. Here we approach this problem through the use of boundary integral equations, an approach first introduced by Kleefeld in [Kl13, Kl15] and one which reduces the dimensionality of the problem by one. For a survey of numerical methods for computing transmission eigenvalues, we refer the reader to Chapter 6 of the recent monograph of Sun and Zhou [SuZo17]. We note that the use of transmission eigenvalues in determining estimates on the constitutive parameters of a medium is restricted to the case when there is no absorption. This is due to the fact that for absorbing media the eigenvalues are complex and hence it is unclear how to determine them from measured data as well as the fact that the abovementioned monotonicity properties are no longer available. The structure of the paper is as follows: first we formulate the problem under consideration in Section 12.2. In Section 12.3 it is explained how the interior transmission eigenvalue problem can be interpreted as a nonlinear eigenvalue problem. Numerical results are given in Section 12.3 for two scattering objects using various parameter choices showing that one is able to compute both real and complex-valued interior transmission eigenvalues. A short summary and an outlook for future work conclude this article.

12.2 Problem Formulation

The problem under consideration is the scattering by an inhomogeneous medium which is given by

us C k2us D 0 in 2nD rAru C k2nu D 0 in D u D us C ui on @D Aru D r.us C ui/ on @D Â Ã @us lim r ikus D 0: r!1 @r

Here, ui is the incident plane wave, us is the scattering wave, denotes the normal pointing in the exterior of the open and bounded domain D. The parameters n 2 and A 2 22 with A symmetric and positive deﬁnite represent the constitutive parameters of the inhomogeneous medium. The abovementioned problem is closely related to the following interior transmission eigenvalue problem 12 Interior Transmission Eigenvalues 141

v C k2v D 0 in D rArw C k2nw D 0 in D w D v on @D Arw D rv on @D :

Values for k 2 ¼ for which the interior transmission eigenvalue problem has a nontrivial solution are called interior transmission eigenvalues. The existence of interior transmission eigenvalues has been studied in [CaCoHa17]. From the theoretical point of view one is interested in those values, since the far-ﬁeld operator has dense range if and only if there does not exist a nontrivial solution to the interior transmission eigenvalue problem. However, the numerical calculation of them is a nontrivial task due to the fact that the problem is nonlinear, non-selfadjoint, and non-elliptic. We will reformulate the problem as a nonlinear eigenvalue problem via boundary integral equations and solve it via complex-valued contour integrals.

12.3 Boundary Integral Equations

In this section, we can closely follow the analysis of Colton et. al [CoKrMo97]. If u is a solution to the Helmholtz equation u C k2u D 0, then uQ.x/ D u.A1=2x/ is a solution of

rArQu C k2uQ D 0 (12.1) as shown in [CoKrMo97, Lemma 2.1]. Hence, the fundamental solution of (12.1)is ˚Q . ; / ˚ . 1=2 ; 1=2 /= 1=2 ˚ . ; / i .1/. / given by k x y D k A x A y det A , where k x y D 4 H0 kjxyj , 2 .1/ x ¤ y is the fundamental solution of u C k u D 0 in two dimensions. Here, H0 is the Hankel function of the ﬁrst kind of order zero. Using Green’s representation formula one obtains Z . / ˚Q p . / . / . / . / ˚Q p . /; w x D k n y Arw y w y y Ary k n ds y x 2 D (12.2) @D is a solution of rArw C k2nw D 0 for w 2 C2.D/ \ C1.D/ and likewise Z

v.x/ D ˚k .y/rv.y/ v.y/ .y/ry˚k ds.y/; x 2 D (12.3) @D is a solution of v C k2v D 0 for v 2 C2.D/ \ C1.D/. Hence, letting the point x 2 D approach the boundary and using the jump relation [CoKrMo97, Lemma 2.2 and Lemma 2.3] yields 142 A. Kleefeld and D. Colton

1 w SQ p Œ A w KQ p Œw w D k n r k n C 2 1 v S Œ v K Œv v D k r k C 2 which gives Á Á Q p Œ Q p Œ 0; Sk n Sk a Kk n Kk b D (12.4) since we have used the boundary conditions a WD Arw D rv and b WD w D v. Here, the deﬁnitions of the boundary integral operators are Z SQk' .x/ WD ˚Qk.x; y/'.y/ ds.y/; x 2 @D ; @ Z D KQ k' .x/ WD .y/ Ary˚Qk.x; y/'.y/ ds.y/; x 2 @D ; @D

with the obvious deﬁnition for Sk and Kk.Using Arw with w taken from (12.2) and similarly v with v taken from (12.3), let the point approach the boundary, and use the jump relation from [CoKrMo97, Lemma 2.2 and Lemma 2.3] gives 1 Arw D KQ 0 p Œ Arw C Arw NQ p Œw k n 2 k n 1 v K0 Œ v v T Œv D k r C 2 r k which gives Á Á Q 0 p 0 Œ Q p Œ 0; Kk n Kk a Tk n Tk b D (12.5) since we have used the boundary conditions a WD Arw D rv and b WD w D v. Here, the deﬁnitions of the boundary integral operators are Z Q 0 ' . / . / ˚Q . ; /'. / . /; @ ; Kk x WD y Arx k x y y ds y x 2 D @ D Z TQk' .x/ WD .x/ Ar .y/ Ary˚Qk.x; y/'.y/ ds.y/; x 2 @D ; @D

0 with the obvious deﬁnition for Kk and Tk. Note that equations (12.4) and (12.5) can be written as 12 Interior Transmission Eigenvalues 143

" # Ä Ä p p SQ Sk KQ C Kk a 0 k n k n D (12.6) KQ 0 p K0 TQ p T 0 k n k k n C k b or abstractly as Z.k/v D 0 with the obvious definitions of Z and v for given n and A. The nonlinear eigenvalue problem is discretized as in Kleefeld [Kl13] and solved with complex-contour integrals as illustrated in Beyn [Be12]. More precisely, each integral operator in the block matrix given in (12.6)is discretized as follows: first, the boundary is subdivided into n parts. Hence, we have T n v a triangulation n D[iD1 i with a total of n points i. Then, the integrand is approximated via quadratic interpolation on each part and collocated with the same points vi. In total, this yields the discretized version of (12.6), that is a nonlinear system of the form Z.k/vE D 0 with Z.k/ 2 ¼2n2n. Note that each entry of this matrix is given by an integral which has been approximated appropriately. This nonlinear system is then solved as follows: first one specifies a contour in the complex plane that may contain interior transmission eigenvalues. We usually choose a circle, say @˝. Next, the two contour integrals with random VO 2 ¼2n`, ` Ä 2n Z Z 1 1 A D Z.k/1VO dk and B D k Z.k/1VO dk 2 i @˝ 2 i @˝ are approximated by the trapezoidal rule yielding

1 NX1 1 XN1 A D Z. .t //1VO 0.t / and B D Z. .t //1VO .t / 0.t /; N iN j j N iN j j j jD0 jD0 where we choose N D 50. Next, we compute a singular value decomposition of the H form AN D V˙W . Then, we perform a rank test for ˙ and ﬁnd 0

1 ::: m > tolrank >mC1 0 ` 0:

If m D `, then one increases ` and starts from the beginning. Otherwise, we let V0 D V.1 W 2n;1 W m/, W0 D W.1 W `;1 W m/, and ˙0 D diag.0;:::;m/ .Next, H 1 mm we compute C D V0 BN W0˙0 2 ¼ : Finally, we solve the linear eigenvalue problem for C. The resulting m eigenvalues are the ones that we are looking for.

12.4 Numerical Results

In this section, we compute the interior transmission eigenvalues for a unit circle C and for an ellipse E with semi-axis a D 1 and b D 0:8. The boundaries of the two scattering objects are shown in Figure 12.1. 144 A. Kleefeld and D. Colton

Fig. 12.1 The boundaries of 2 the two-dimensional scatterers: the unit circle and the ellipse with semi-axis 1 a D 1 and b D 0:8

-1

-2 -2 -1 0 1 2

-1

-2 -2 -1 0 1 2

Table 12.1 The ﬁrst ﬁve ITP C E real-valued interior transmission eigenvalues for a 1 2.903 3.135 circleandanellipseusingthe 2 2.903 3.485 parameters n D 4 and A D I 3 3.384 3.547 4 3.412 3.884 5 3.412 4.146

First, we use the index of refraction n D 4 and the matrix A is given by the identity. We are able to calculate the first five real-valued interior transmission eigenvalues (ITP) for the unit circle and the ellipse. The values are presented in Table 12.1. The results for the two scatterers are in agreement with the results of Cakoni &Kress[CaKr16, Table 4.1]. Interestingly, we are able to compute complex- valued interior transmission eigenvalues although the existence of those from the theoretical point of view is still open. We obtain, for example, 3:974 ˙ 0:567ifor the unit circle and 4:205 ˙ 0:573i for the ellipse. Next, we compute the first four 12 Interior Transmission Eigenvalues 145

Table 12.2 The ﬁrst four ITP C E real-valued interior transmission eigenvalues for a 1 2.921 3.155 circleandanellipseusingthe 2 2.924 3.512 parameters n D 4 and 3 3.410 3.574 A D diag.1:01; 1:01/ 4 3.430 3.902

Table 12.3 The ﬁrst four ITP C E real-valued interior transmission eigenvalues for a 1 2.909 3.144 circleandanellipseusingthe 2 2.919 3.504 parameters n D 4 and 3 3.394 3.552 A D diag.1; 1:01/ 4 3.423 3.899

Table 12.4 The ﬁrst four ITP C E real-valued interior transmission eigenvalues for a 1 2.903 3.144 circleandanellipseusingthe 2 2.926 3.516 parameters n D 4 and 3 3.386 3.552 A D Œ1; 0:01I 0:01; 1:01 4 3.423 3.901 interior transmission eigenvalues for n D 4 and A D diag.1:01; 1:01/. The values for the unit circle and the ellipse are given in Table 12.2. As we can see, the interior transmission eigenvalues given in Table 12.2 are slightly larger than the values given in Table 12.1. More precisely, we have a monotonicity behavior for the interior transmission eigenvalues for increasing a0 with A D a0I for n D 4. This observation is consistent with the theoretical results on the monotonicity of transmission eigenvalues given in [CaCoHa17]. Again, we are also able to compute complex-valued interior transmission eigenvalues. We get 3:981 C 0:554i for the unit circle and 4:208 C 0:562i for the ellipse. Now, we consider the parameters n D 4 and A D diag.1; 1:01/. The interior transmission eigenvalues for the unit circle and the ellipse are presented in Table 12.3. Interestingly, the values in Table 12.3 are slightly smaller than the values given in Table 12.2 and slightly larger than the values given in Table 12.1 which shows again the monotonicity. It should be pointed out that this observation is again consistent with the theoretical results on the monotonicity of transmission eigenvalues given in [CaCoHa17]. We are able to compute the complex-valued interior transmission eigenvalues 3:976 C 0:565i and 4:203 C 0:572i for the unit circle and the ellipse, respectively. Last, but not least, we use n D 4 and the symmetric positive deﬁnite matrix A D Œ1; 0:01I 0:01; 1:01. The acquired values for the unit circle and the ellipse are given in Table 12.4. We obtain the complex-valued interior transmission eigenvalues 3:973 C 0:572i and 4:202 C 0:572i for the unit circle and the ellipse, respectively. 146 A. Kleefeld and D. Colton

Table 12.5 The ﬁrst four ITP C E real-valued interior transmission eigenvalues for a 1 2.485 2.616 circleandanellipseusingthe 2 2.991 2.962 parameters n D 4 and 3 3.510 3.474 A D diag.1=5; 1=5/ 4 4.028 3.803

Finally, we present numerical results for the unit circle and for the ellipse in Table 12.5 using the parameters n D 1 and A D diag.1=5; 1=5/. We obtain the complex-valued interior transmission eigenvalues 3:821 C 0:573i and 3:967 C 0:572i for the unit circle and the ellipse, respectively.

12.5 Summary and Outlook

In this paper, we are able to show that we can compute interior transmission eigenvalues for anisotropic media for various obstacles in two dimensions using boundary integral equations and a nonlinear eigenvalue solver based on complex- valued contour integrals. The results can be easily extended for obstacles in three dimensions which we will consider in a future article also providing more details. Additionally, it remains to show that there are complex-valued interior transmission eigenvalues although we were able to show numerically that they exist. The ﬁnal step is the extension of this work to the electromagnetic and elastic scattering problem. It would also be interesting to investigate whether one is able to calculate the interior transmission eigenvalues for anisotropic media with the inside-outside duality method (see original work by Kirsch & Lechleiter [KiLe13] and the work of [LePe15a, LePe15b, LeRe15, PeKl16]), since we are now able to calculate far- ﬁeld data for it. From the theoretical point of view is not known if the inside-outside duality method will work or not.

Acknowledgements The research of the second author was supported in part by a grant from the United States Air Force Ofﬁce of Scientiﬁc Research.

References

[Be12] Beyn, W.-J.: An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl. 436, 3839–3863 (2012) [CaKr16] Cakoni, F., Kress, R.: A boundary integral equation method for the transmission eigenvalue problem. Appl. Anal. 96, 23–38 (2017) [CaEtAl10] Cakoni, F., Colton, D., Monk, P., Sun, J.: The inverse electromagnetic scattering problem for anisotropic media. Inverse Prob. 26, 074004 (2010) 12 Interior Transmission Eigenvalues 147

[CaCoHa17] Cakoni, F., Colton, D., Haddar, H.: Inverse Scattering Theory and Transmission Eigenvalues. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 88. SIAM, Philadelphia (2017) [CoKrMo97] Colton, D., Kress, R., Monk, P.: Inverse scattering from an orthotropic medium. J. Comput. Appl. Math. 81, 269–298 (1997) [KiLe13] Kirsch, A., Lechleiter, A.: The inside-outside duality for scattering problems by inhomogeneous media. Inverse Prob. 29(10), 104011 (2013) [Kl13] Kleefeld, A.: A numerical method to compute interior transmission eigenvalues. Inverse Prob. 29, 104012 (2013) [Kl15] Kleefeld, A.: Numerical methods for acoustic and electromagnetic scattering: Transmission boundary-value problems, interior transmission eigenvalues, and the factorization method. Habilitation thesis, Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus (2015) [LePe15a] Lechleiter, A., Peters, S.: The inside-outside duality for inverse scattering problems with near ﬁeld data. Inverse Prob. 31(8), 085004 (2015) [LePe15b] Lechleiter, A., Peters, S.: Determining transmission eigenvalues of anisotropic inhomogeneous media from far ﬁeld data. Commun. Math. Sci. 13(7), 1803–1827 (2015) [LeRe15] Lechleiter, A., Rennoch, M.: Inside-outside duality and the determination of electromagnetic interior transmission eigenvalues. SIAM J. Math. Anal. 47(1), 684–705 (2015) [MoSu12] Monk, P., Sun, J.: Finite element methods for Maxwell transmission eigenvalues. SIAM J. Sci. Comput. 34, B249–B264 (2012) [PeKl16] Peters, S., Kleefeld, A.: Numerical computations of interior transmission eigenvalues for scattering objects with cavities. Inverse Prob. 32(4), 045001 (2016) [SuZo17] Sun, J., Zhou, A.: Finite Element Methods for Eigenvalues Problems. CRC Press, Boca Raton (2017) Chapter 13 Improvement of the Inside-Outside Duality Method

A. Kleefeld and E. Reichwein

13.1 Introduction and Motivation

The interior transmission eigenvalue problem appears initially in inverse acoustic scattering theory. Kirsch in 1986 and Colton & Monk in 1988 were the first to mention this problem in [Ki86] and [CoMo88], respectively. Furthermore, the existence of real interior eigenvalues [CaHaGi10] and their discreteness [Ki86] has already been shown, but the existence of complex valued interior eigenvalues of arbitrary obstacle is up to this day an open question. There are still a lot of unresolved issues regarding this theory and it is an active field of research in theoretical as well as in practical regards (see [CaCoHa17] for a recent survey). Interior transmission eigenvalues can be used to find abnormalities inside homogeneous obstacles. This is attainable, since they satisfy a monotonicity principle [CaHa13] and the eigenvalues of homogeneous objects and objects with inhomogeneities differ from each other. With this it is even possible to tell the size and the location of the inhomogeneities by knowing the interior transmission eigenvalues. It is therefore desirable to compute highly accurate eigenvalues in order to find these inhomogeneities inside an obstacle. The numerical calculations of these eigenvalues is challenging, since the interior transmission problem is neither elliptic nor selfadjoint. Up to now, there are only a few methods available to calculate interior transmission eigenvalues like the method of Cossonniére [Co11], the method of Kleefeld [Kl13, Kl15b, Kl15a], or the inside-outside duality method [KiLe13],

A. Kleefeld () Supercomputing Centre, Forschungszentrum Jülich GmbH, Jülich, Germany e-mail: [email protected] E. Reichwein Faculty of Mathematics and Natural Sciences, Heinrich-Heine-Universität, Düsseldorf, Germany e-mail: [email protected]

© Springer International Publishing AG 2017 149 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_13 150 A. Kleefeld and E. Reichwein and others like [CaKr16, JiSu13, JiSuXi14, LiEtAl15, MoSu12, SuXu13, SuZh17, ZeSuXu16]. But they either need too much information about the scatterer’s surface like the method of Kleefeld or are too inaccurate and expensive like the inside- outside duality method or the method of Cossonniére. While we want to improve the accuracy of interior transmission eigenvalues without knowing the scatterer’s surface, this paper is about improving the inside-outside duality approach, since this method links interior transmission eigenvalues and the far-ﬁeld data of the corresponding scattering problem.

13.2 Problem Formulation

As mentioned in the introduction, the interior transmission eigenvalue problem arises in scattering theory, which is why we introduce the scattering problem first. After that we describe the interior transmission problem to get a link between both. For that purpose let D 3 be the scatterer with boundary @D, which is defined by a function n 2 L1.D/, such that n >1inside of D and n D 1 outside of D. We consider the following scattering problem: for an arbitrary wave i 2 3 u .x/ D expfix dOg with direction dO 2 Dfx 2 WjxjD1g and wavenumber we seek the total field u , which satisfies

2 u C nu D 0

3 s i in , such that the scattering ﬁeld u D u u fulﬁlls Sommerfeld’s radiation condition, i.e.

s s 2 @ru .x/ iu .x/ D O.r /; r Djxj!1:

s The scattering ﬁeld u has the asymptotic behavior

ir e 3 us .x/ D u1.xOI dO/ C O.r 2 /; 4 r

2 1 where r converges uniformly to infinity in xO D x=jxj2 and u is the far-field s pattern of the scattering field u . The wavenumber is called an interior acoustic transmission eigenvalue, if non-trivial functions v;w 2 L2.D/; v w 2 H2.D/ \ 1 H0.D/ exist, which solve the interior transmission eigenvalue problem

w C 2nw D 0 in D ; v C 2v D 0 in D ; w D v on @D ; @ @ v @ : Ew D E on D 13 Improvement of the Inside-Outside Duality Method 151

Here, E denotes the normal pointing in the exterior. The far-ﬁeld operator FW L2.2/ ! L2.2/ is deﬁned through Z 1 .F g/.xO/ D u .xOI dO/g.dO/ d.dO/; 2 which is normal and compact. The eigenvalues .p; /p2 of F are lying on a circle in the complex plane with radius 8 2= and center-point 8i 2= (see [KiLe13]). If we represent each p; in polar coordinates

ip; p; Djp; je ;p; 2 Œ0; /; every p; gets a corresponding phase p; .Forp; D 0 we set p; D 0.Itis known that p; ! 0 from the right for p !1.ItexistsaL with the maximal phase L D maxp2 p; . If there exist an wavenumber 0, so that L ! for ! 0, then 0 is an interior transmission eigenvalue [KiLe13]. In other words, the behavior of the eigenvalues of F characterizes the interior transmission eigenvalues, which is an essential part of the inside-outside duality method.

13.3 The Inside-Outside Duality Method

The calculation of interior transmission eigenvalues with the inside-outside duality method goes back to Kirsch & Lechleiter [KiLe13] and co-authors [LePe15a, LePe15b, LeRe15]. To obtain these eigenvalues, we first have to generate a fine- meshed, equidistant grid for the wavenumber . After that we approximate the far-field operator F as follows: we split the unit sphere in N parts of equal size, e.g., N D 120 (this strategy has been used in [JiLe15, LePe14, LePe15a, LePe15b, PeKl16]). Each part has the surface area ! D 4 =N, where 4 is the size of the unit sphere’s surface. Then the integrand of each part is approximated by constant interpolation. The center-points of each section are generated with the algorithm of Cessenat (see [Ce96]) and represented by xO`;`D 1;:::;N. We obtain

XN 1 .F g/.xO`/ !u .xO`I dOj/g.dOj/ jD1 2 3 2 3 1 1 1 u .xO1I dO1/ u .xO1I dO2/ ::: u .xO2I dON / g.dO1/ 6 1 1 1 7 6 7 6u .xO2I dO1/ u .xO2I dO2/ ::: u .xO1I dON /7 6g.dO2/7 D ! 6 : : : 7 6 : 7 : 4 : : : 5 4 : 5 1 1 1 u .xO I dO1/ u .xO I dO2/ ::: u .xO I dO / g.dO / „ N ƒ‚N N N … N N F 152 A. Kleefeld and E. Reichwein

N N We compute the eigenvalues p; of F as an approximation of the eigenvalues p; of the far-ﬁeld operator F for each grid-point . Now, we investigate the N behavior of the corresponding phases p; with varying wavenumber. To obtain the interior transmission eigenvalues, we have to examine the wavenumbers 0 for N which the maximum phase L p; converges to for ! 0. Note that we need a highly accurate approximation of F to get highly accurate interior transmission eigenvalues. For that purpose we need an N which is larger than 120 or alternatively a better approximation. But a large N makes this method very expensive. That leads us to the question: Are there better ways to approximate the far-ﬁeld operator F ?

13.4 Improvement and Numerical Results

To get a better approximation of the far-ﬁeld operator,R we use different kinds of 1 interpolations for the integral operator .F g/.xO/ D 2 u .xOI dO/g.dO/ d.dO/,more precisely the product Gaussian quadrature [At82], the Lebedev quadratures ([Be15, p. 4–5]), and the spherical t-Design ([Be15, p. 5–7]). Because it is needed in the ˘ K m.; / 0 ; following sections, we will deﬁne D spanfYn W Ä n Ä K n Ä m Ä ng as the subset of spherical harmonics up to degree K with polar coordinates 2 Œ0; 2 / and 2 Œ0; .

Product Gaussian Quadrature

A different way is shown by Atkinson [At82]. To approximate the far-ﬁeld operator with this ansatz, we rewrite F g in spherical coordinates,

Z Z 2 Z 1 1 .F g/.xO/ D u .xOI dO/g.dO/ d.dO/ D U .xOI ; /G.; / sin./ d d ; 2 0 0

1 1 with U .xOI ; / D u .xOI dO/ and G.; / D g.dO/. Then the latter integral is approximated by

X2m Xm 1 .F g/.xO`/ ! u .xO`I dO /g.dO /: m i ij ij jD1 iD1

2 T Each xO` 2 is deﬁned by xO` D dOij D Œsin.i/ cos. j/; sin.i/ sin. j/; cos.i/ , 2 ` D 1;:::;2m ; j Dd`=me; i D ` .j 1/m: The points j are equally spread on Œ0; 2 with distance =m.Theyaresetby j D j=m: The fig are chosen, such that 13 Improvement of the Inside-Outside Duality Method 153 fcos.i/g are the Gauss-Legendre nodes on Œ1; 1 with the corresponding weights !i. With this choice of node points, the product Gaussian quadrature integrates exactly every polynomial of degree less than 2m. The numerical values for the nodes and weights have been collected from [HaTo13]. As a consequence, 2 3 2 3 v 1. / ::: v 1. / . / 1 u xO1I dO1 2m2 u xO1I dO2m2 g dO1 6 v 1. / ::: v 1. / 7 6 . / 7 6 1 u xO2I dO1 2m2 u xO2I dO2m2 7 6 g dO2 7 .F g/.xO`/ 6 : : 7 6 : 7 ; m 4 : : 5 4 : 5 1 1 v1 u .xO2 2 I dO1/ ::: v2 2 u .xO2 2 I dO2 2 / g.dO2 2 / „ m ƒ‚m m m … m N F (13.1)

2 where v` D !i with ` D 1;:::;N; i D ` .d`=me1/m, and N D 2m .

13.4.1 Lebedev Quadrature

The Lebedev quadrature is deﬁned as follows. We rewrite as before F g in spherical coordinates,

Z Z 2 Z 1 1 .F g/.xO/ D u .xOI dO/g.dO/ d.dO/ D U .xOI ; /G.; / sin./ d d 2 0 0 to obtain a quadrature scheme, which uses the quadrature-rules for one-dimensional integrals twice. Therefore we get, similar to the case of the product Gaussian quadrature,

XL XM XL XM 1 1 .F g/.xO/ !i vju .xOI i; j/g.i; j/ D !Qiju .xOI i; j/g.i; j/: iD1 jD1 iD1 jD1

N 2 One then obtains the approximation F of the far-ﬁeld operator for xO` 2 ; l D 1;:::;N and N D LM is analogously to (13.1) only with different dimensions and different weights and coordinates. To calculate the latter, Lebedev constructed a class of quadratures, which is invariant under the octahedron rotation group by working out the invariant spherical harmonics and solving the non-linear equations of the spherical harmonics for degrees up to order p. The idea behind this scheme is to integrate spherical harmonics up to a certain order by reducing the number of non-linear equations that need to be solved. In our case, the variable M is equal to 154 A. Kleefeld and E. Reichwein

L and the product of both is the degree, which is given by N D p.p C 1/=3: The construction of the Lebedev quadrature can be a tedious task, especially without pre-computed data, because a large number of non-linear equations have to be solved simultaneously. This makes this scheme less adaptive than the product Gaussian quadrature. However, it is more efﬁcient than the Gaussian scheme, one can nevertheless obtain the same kind of accuracy using less quadrature points (see [Be15, p. 4–5]). Consequently, we get a faster integration scheme if the nodes and weights are pre-computed. Those numerical values for the coordinates and weights have been collected from Lebedev [LeLa99], which derived data up to order p D 131 (N D 5810/.

13.4.2 Spherical t-Design

This interpolation was initially deﬁned by Delsarte, Goethals, and Seidel as a problem in algebraic combinatorics (see [DeGoSe77]). This quadrature is based on the idea to integrate all spherical harmonics up to a ﬁxed order p. It assumes that all weights for the N interpolation points are equal. Since the scheme is supposed to integrate the constant function f Á 1 exactly, we receive the following weights

!i D 4 =N ; i D 1;:::;N :

This rises the question of how the nodes should be spread around the sphere. Because all weights are equal, the nodes should be spaced consistently over the sphere. Moreover, a set of N points fxig is known as a spherical t-design, if it satisﬁes Z 4 XN t f .x/ d˝ D f .xi/; 8f 2 ˘ : 2 N iD1

Obviously, one wants to create a spherical design with a minimal number of nodes to keep the computational time small. To this end, the existence of efﬁcient spherical design for arbitrary degrees was already demonstrated in 1984, but there was no N evidence of how many points were used in this design. To get an approximation F 2 of the far-ﬁeld operator F for some points xO` 2 ;`D 1;:::;N, we put this weights and nodes in formula (13.1). Therefore, we receive 2 3 2 3 1 1 1 u .xO1I dO1/ u .xO1I dO2/ ::: u .xO1I dON / g.dO1/ 6 1 1 1 7 6 7 4 6u .xO2I dO1/ u .xO2I dO2/ ::: u .xO2I dON /7 6g.dO2/7 .F g/.xO/ 6 : : : 7 6 : 7 : N 4 : : : 5 4 : 5 1 1 1 u .xO I dO1/ u .xO I dO2/ ::: u .xO I dO / g.dO / „ N ƒ‚N N N … N N F 13 Improvement of the Inside-Outside Duality Method 155

For the computation of the nodes of the spherical designs xO` D dO`;`D 1;:::;N we refer to Burkardt [ZhEtAl16], which contains a dataset that derived spherical designs up to order p D 21 (N D 240).

13.4.3 Numerical Results

We compare four different methods to approximate the far-ﬁeld operator, which were presented in the previous sections, to see whether they have or have not an effect on the accuracy of the numerical calculated phases. Thus, we get also the impact of the accuracy of the interior transmission eigenvalues by the interpolation methods. As mentioned before, we use nodes and weights for the product Gaussian quadrature [HaTo13], the Lebedev quadrature [LeLa99], and the spherical t-design [ZhEtAl16]. We make use of the far-ﬁeld of the transmission problem for a sphere of radius R [AnChKl13], which is given by p p p 4 X1 nj0 . nR/j .kR/ j0 .kR/j . nR/ 1. O/ i .2 1/ p p p p . O/; u xOI d D p C p p .1/ .1/0 Pp xO d 0 . / . / . / . / pD0 n jp n R hp kR hp kR jp kR with exact eigenvalues p p p 16 2 nj0 . nR/j .R/ j . nR/j0 .R/ p p p p ; p; D p .1/0 p .1/ p i . / . / . / 0 . / jp n R hp R nhp R jp n R

.1/ where jp denotes the ﬁrst kind spherical Bessel-function, hp the spherical Hankel- function of the ﬁrst kind, n the index of refraction, and Pp the Legendre-polynomial. Consequently, we get the exact phases Â Ã 1 p; p; D log : i jp; j

With the methods mentioned above, we calculate the relative error between the numerical phase for p D 1; 2; 3 and the exact phase for p D 1; 2; 3 for different degrees, wavenumbers D 2 and D 3, n D 4, and R D 1, see Figures 13.1 (A) and (B). One can see that the relative error is much larger by using constant interpolation than it is with less degree by using product Gaussian quadrature, Lebedev quadrature, or spherical t-design. Especially the Lebedev quadrature calculates highly accurate phases. We plot the numerical phases calculated with Lebedev quadrature (N = 86) against the wavenumbers 2 Œ2; 6:5, see Figure 13.2. 156 A. Kleefeld and E. Reichwein

(A) Wavenumber κ = 2 (B) Wavenumber κ = 3 10-2 10-2

10-4 10-4

10-6 10-6

10-8 10-8

10-10 10-10 Relative error Relative error

10-12 10-12

10-14 10-14

10-16 10-16

10-18 10-18 101 102 103 101 102 103 N N

Fig. 13.1 Relative error between the numerical phases and the exact phases with wavenumbers D 2 (plot A) and D 3 (plot B) for the transmission problem and for various quadratures using a sphere with radius R D 1 and index of refraction n D 4. The constant interpolation is characterized by diamonds, the product Gaussian quadrature by circles, the Lebedev quadrature by triangles, and the spherical t-design approximation by squares. The black markers denote the relative error for 1, the grey markers for 2, and the white ones for 3

Usually the inside-outside duality approach is used with constant interpolation and degree N D 120, which is slower in computational time than Lebedev quadrature with N D 86. The exact interior transmission eigenvalues in this interval are D 3:141593, 3:692445, 4:261683, 4:831855, 5:399436, 5:963690, 6:283185, and 6:373336. Using Lebedev quadrature with N D 86, we obtain the above calculated interior transmission eigenvalues accurate within the chosen grid size. We obtain the same results using near-ﬁeld data (see also [LePe15a, p. 17]). As a remark, 13 Improvement of the Inside-Outside Duality Method 157

Lebedev quadrature N = 86 3.5

2.5

μ 2

1.5 Phase

0.5

0 22.533.544.555.566.5 Wavenumber κ

Fig. 13.2 Detection of eight interior transmission eigenvalues of the sphere with radius R D 1 and index of refraction n D 4 using Lebedev quadrature with N D 86 we would like to mention that if we use only the data of the upper half-sphere in the different approximations of the far-ﬁeld operator, our resulting phases are not even close to the exact phases. Thus, it is not possible to calculate interior transmission eigenvalues with these limited data. Of course, we are also able to calculate interior Dirichlet eigenvalues. Using Lebedev with N D 84 yields 3:14, 4:49, 5:23, and 5:76 for the unit sphere and 2:97, 4:05, 4:35, and 5:16 for the prolate spheroid with parameters b D 6=5 and a D 1, respectively. The results are accurate within the chosen grid size of 0.01 (compare with [Kl13, Table 11]). The results for the constant interpolation are not that accurate. For example, we obtain 5:17 instead of 5:16 for the prolate spheroid.

13.5 Summary and Outlook

We considered four different possibilities to approximate the far-ﬁeld integral operator. With these we illustrated that the common way of the inside-outside duality with constant interpolation (used in [JiLe15, LePe14, LePe15a, LePe15b, PeKl16]) computes less accurate interior transmission eigenvalues than, for example, the Lebedev quadrature. To receive highly accurate interior transmission eigenvalues, the inside-outside duality should be improved by using a more accurate interpolation than the constant interpolation especially for larger wavenumbers. As future work, we try to construct an adaptive mesh for the wavenumber to avoid large computation time. Additionally, we also want to investigate a possible improvement for the calculation of electromagnetic interior transmission eigenvalues. 158 A. Kleefeld and E. Reichwein

References

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P.A. Kozarzewski and E. Zappale

14.1 Introduction

Optimal design problems, devoted to find the minimal energy configurations of a mixture of two conductive (or elastic) materials, have requested much attention in the past years starting with the pioneering papers [KoEtAl86]. It is well known that, given a container ˝ and prescribing only the volume fraction of the material where it is expected to have a certain conductivity, an optimal configuration might not exist. To overcome this difficulty, Ambrosio and Buttazzo in [AmEtAl93] imposed a perimeter penalization on the interface of the two materials. In [CaEtAl12,formula (2)] the same perimeter term has been added in order to deal with the model proposed in [FoEtAl98] and [BrEtAl00] in the framework of thin structures in the nonlinear elasticity setting. Here we are considering an analogous problem, where the continuous energy 33 densities Wi W ! , i D 1; 2, do not satisfy growth conditions of order p but are of the type

0 ˇ.˚i.jFj/ 1/ Ä Wi.F/ Ä ˇ .1 C ˚i.jFj// (14.1)

33 for every F 2 and ˚i – equivalent Orlicz functions (see Section 14.2) with 0<ˇÄ ˇ0. We refer to [LaEtAl13, LaEtAl14] for related results in the framework of dimensional reduction problems casted in the Orlicz-Sobolev setting.

P.A. Kozarzewski () University of Warsaw, Warsaw, Poland Military University of Technology, Warsaw, Poland e-mail: [email protected] E. Zappale Department of Industrial Engineering, University of Salerno, Fisciano, SA, Italy e-mail: [email protected]

Let ">0and consider ˝."/ WD ! ."; "/, where ! is a bounded domain of 2, with Lipschitz boundary. Assume that ˝."/ is clamped on its lateral boundary, and suppose that ˝."/ is filled with two materials of respective energy densities Wi, i D 1; 2 as above, satisfying (14.1). We study the following minimum problem Z 1 inf " Œ.E."/W1 C .1 E."//W2/.rv/ C f vdx v 2 W1;˚ .˝."/I 3/ ˝."/ .˝."/ 0; 1 / E."/ 2 BV If g Z 1 . ."/ ˝."// v 0; 1 ; C " P E I W b@!.";"/D ˝."/ E."/ dx D j j ˝."/ (14.2) where E ."/ ˝ ."/ is a measurable set with finite perimeter, ˚ is a function ˚ 3 equivalent to ˚i, i D 1; 2, and fN 2 L ˝."/I , where ˚ is the Hölder conjugate of ˚ (see Section 14.2) and 2 .0; 1/ is the volume fraction. In order to study the asymptotic behaviour of (14.2) we first rescale the problem in a fixed 3D domain and then we perform convergence with respect to the pair (deformation, design region) as in [CaEtAl12]. We refer to [Da93]for 1 : -convergence theory. We consider a " dilation in the transverse direction x3 Set ˝ WD ! .1; 1/ ;

E" WD f.x˛; x3/ 2 ˝ W .x˛;"x3/ 2 E ."/g ; u .x˛; x3/ WD v .x˛;"x3/ ; . ; / . ;" / ;. ; / . ;" / ; (14.3) f x˛ x3 WD f x˛ x3 E" x˛ x3 WD E."/ x˛ x3 where x˛ WD .x1; x2/; and v is any admissible field for (14.2). In the sequel we will denote dx˛ WD dx1dx2 and r˛ and D˛ will be identified 32 with the pair .r1; r2/ ; .D1; D2/ ; respectively. For every matrix F 2 and any z 2 3, we denote by F WD .Fjz/ the matrix in 33 whose first two columns are those of F and the last column is given by the vector z. ˇ ˇ ˇ ˇ By the definition of total variation, P .E ."/ I ˝ ."// D DE."/ .˝ ."//, and the change of variables in (14.3) ensures that ˇ ˇ ˇ ˇ 1 ˇ ˇ .˝."// ˇ ; 1 ˇ .˝/; " D E."/ D D˛ " " D3 " where E" denotes the characteristic function of E", that in the sequel we will indicate simply by ". We refer to [AmEtAl00] for sets of finite perimeter and BV functions. 1 ˚ 3 For every ">0,letJ" W L .˝If0; 1g/L .˝I / ! Œ0; C1 be the functional defined by 8 R ˇ R ˇ 1 ˆ .W1 C .1 /W2/ r˛u r3u dx C f udx <ˆ ˝ˇ ˇ ˇ " ˝ ˇ ˇ 1 ˇ .˝/ .˝ 0; 1 / 1;˚ .˝ 3/; .; / C D˛ " D3 in BV If g W I J" u WD ˆ :ˆ C1 otherwise. (14.4) 14 Optimal Design in Orlicz–Sobolev Setting 163

1 ˚ 3 Similarly, consider the functional J0 W L .˝If0; 1g/ L .˝I / ! Œ0; C1 deﬁned by

8 R R 1 R ˆ 2 QV.; r˛u/dx˛ C f udx˛dx3 <ˆ ! 1 ! 2 .!/; .! 0; 1 / 1;˚ .! 3/; .; / C jD j in BV If g W I J0 u WD ˆ (14.5) :ˆ C1 otherwise, where V Wf0; 1g33 ! Œ0; C1/ is given by

V.d; F/ WD dW1.F/ C .1 d/W2.F/; (14.6)

32 with W1 and W2 satisfying (14.1), V Wf0; 1g ! Œ0; C1/ is given by V d; F WD dW1 F C .1 d/ W2 F ; (14.7) . / ; 32; 1; 2; with Wi F WD infc23 Wi Fjc F 2 i D and QV stands for the quasiconvexiﬁcation of V in the second variable. Namely, for every .d; F/ 2 f0; 1g32 Z 1 0 3 QV d; F WD inf V d; F Cr˛'.x˛/ dx˛ W ' 2 C0 Q I ; (14.8) Q0 where Q0 2 denotes the unit cube. We will prove that problems (14.2) converge, " ! 0C, to the problem

Z Z 1 Z inf 2 QV.; r˛u/dx˛ f udx C 2jDj.!/ : 1;˚ 3 ! 1 ! u 2 W0 .!I / .! 0; 1 / 2RBV If g 1 1 j!j ! dx˛ D 2

In fact, the above convergence relies on the following theorem that will be proven in Section 14.3. We underline that the strategy of the proof is similar to the analogous result in [CaEtAl12], but it requires to introduce ad hoc tools in the Orlicz-Sobolev setting. Theorem 1 Let ˝ D ! .1; 1/ be a bounded open set, ! Â 2 open and bounded with Lipschitz boundary and let Wi W ˝ ! Œ0; C1/,i D 1; 2, be continuous functions satisfying (14.1). Let .J"/ be the family of functionals deﬁned in (14.4). Then .J"/-converges, with respect to the strong topology of 1 ˚ 3 C L .˝If0; 1g/ L .˝I / to J0 in (14.5),as" ! 0 . Indeed for what concerns the volume constraint and the boundary conditions, it is enough to observe that the strong convergence in L1.˝If0; 1g/ L˚ .˝I 3/ 164 P.A. Kozarzewski and E. Zappale

(and weak*-BV.˝If0; 1g/ weak-W1;˚ .˝I 3/ see page 7), of the rescaled sequence ."; u"/ of almost minimizers of problem (14.2)to.; u/ 2 1;˚ 3 BV.!If0; 1g/ W0 .!I /, guarantees that the volume fraction Z Z 1 1 E."/dx D "dx D j.˝."/j ˝."/ j˝j ˝ is kept in the limit. The continuity of the trace operator [KaEtAl13, Theorem] entails 1;˚ 3 that u 2 W0 .!I /.

14.2 Preliminaries

We say that ˚ W Œ0; C1/ ! Œ0; C1/ is an Orlicz function whenever it is 0 ˚. /= 0 continuous, strictly increasing, convex, vanishes only at and limt!0C t t D ; Rlimt!C1 ˚.t/=t DC1: This statement is equivalent to demand that ˚.t/ D t 0 .s/ds for some right-continuous, non-decreasing s.t. .t/ D 0 Á t D 0 and lim .t/ DC1: t!C1 We say that ˚ satisﬁes 2 (denoted by ˚ 2 2) condition whenever

there exists C >0and t t0 such that ˚.2t/

Orlicz functions ˚ possess the complementary Orlicz function .s/ WD ˚ .s/, where the latter denotes the standard Fenchel’s conjugate of ˚, i.e.

.s/ WD supfst ˚.t/g; s 0; t0 R s 1 1 and it results that .s/ D 0 ./d; where stands for right inverse function of . Clearly D .˚ / D ˚. If 2 2, then (see [KrEtAl61, Theorem 4.2])

there exists C >0and t0 0 such that ˚.t/ Ä 1=.2C/˚.Ct/ for any t > t0: (14.10) Given two Orlicz functions ˚ and ˚ 0, ˚ dominated ˚ 0 near inﬁnity (˚ 0 ˚ or ˚ ˚ 0 in symbols)

0 if there exists C >1and t0 >0such that ˚ .t/ Ä ˚.Ct/ for all t > t0: (14.11)

We say that Orlicz functions ˚;˚0 are equivalent whenever ˚ ˚ 0 ˚: For an arbitrary set of positive Lebesgue measure E we recall that theR Orlicz class . / ˚. / < L˚ E of functions u on E are the functions satisfying inequality E juj dx ˚ C1: In general L˚ .E/ is not a linear space and the Orlicz space L .E/ is deﬁned as the linear hull of L˚ .E/. On the other hand (see [KrEtAl61, Theorem 8.2]), Orlicz ˚ class L˚ .E/ coincides with its Orlicz space L .E/ if and only if ˚ 2 2: 14 Optimal Design in Orlicz–Sobolev Setting 165

Orlicz spaces are equipped with Luxembourg norm, namely Z

jjujj ˚ . / D inf ˚.juj=k/ Ä 1 (14.12) L E >0 k E and are complete (see [KrEtAl61, Theorems 9.2 and 9.5]). The following properties, that will be used in the sequel, hold (we follow the scheme in [TaEtAl12])

Lemma 1 Let ˚ W Œ0; C1/ ! Œ0; C1/ be an Orlicz function 2 2 and let E be a bounded open set in N . Then (i) C1 is dense in L˚ ([Go82, Theorem 1]); (ii) L˚ .E/ is separable ([KrEtAl61, point 4 at page 85]) and it is reflexive when ˚ satisfies (14.10) [KrEtAl61, Theorem 14.2]; (iii) the dual of L˚ .E/ is identified with L .E/,( D ˚ ) and the dual norm on . / L E is equivalent to kkL [KrEtAl61, Theorem 14.2]; (iv) given ˚ and ˚Q , the continuous embedding L˚ .E/,! L˚Q .E/ holds iff ˚ ˚Q (see (14.11)) near infinity [KrEtAl61, Theorem 8.1]; In particular, whenever 0 ˚;˚0 are equivalent Orlicz functions we have that spaces L˚ ; L˚ coincide and their norms are equivalent. ˚ . /, 1. /, 1 . /, D0. / (v) in view of (iv) L E ! L E ! Lloc E ! E ; Sobolev-Orlicz spaces W1;˚ .E/ are defined as follows:

W1;˚ .E/ WD fu 2 D0.E/ W u 2 L˚ .E/; ru 2 .L˚ .E//dg endowed with the norm

; jjujjW1;˚ .E/ WD jjujjL˚ .E/ C jjrujjL˚ .E/IN

(where the meaning of the norm kkL˚ .EId/ is easily understood from the deﬁnition (14.12)), thus they are Banach spaces. The Sobolev-Orlicz space W1;˚ .EI d/, d 2 is deﬁned as the Banach space of d valued functions u 2 L˚ .EI d/ with distributional derivative ru 2 L˚ .EI Nd/, equipped with the norm

: kukW1;˚ .EId/ WD kukL˚ .EId/ CkrukL˚ .EINd/

Remark 1 In fact [Go82, Theorem 1] deals with density of smooth functions in 1;˚ space W .E/: Indeed, for ˚ 2 2 smooth functions are dense in the space 1;˚ 1;˚ W .E/ and thus W .E/ is separable. Without the assumption of ˚ 2 2 these properties do not hold. 166 P.A. Kozarzewski and E. Zappale

If E has Lipschitz boundary, then the embedding

W1;˚ .EI d/,! L˚ .EI d/ (14.13) is compact (see [Ad77] and [GaEtAl99, Theorems 2.2 and Proposition 2.1]). For Sobolev-Orlicz space W1;˚ .E/; where E has a Lipschitz boundary we have a Poicaré inequality (see [DoEtAl71, Theorem 3.4 (a)])

. ;˚/>0 jjujjL˚ .E/ Ä CjjrujjL˚ .E/ for some constant C D C E

1;˚ and if ˚ 2 2, there exists a linear continuous trace operator Tr W W .E/ ! L˚ .@E/ [KaEtAl13, Theorem 3.13]. The following result, whose proof is immediate, will be exploited in the sequel Proposition 1 Let Vbeasin. 14.7/ : Then V is continuous and satisﬁes ˇ ˇ ˇ ˇ ˇ0 ˚.ˇFˇ/ 1 Ä V ; F Ä ˇ 1 C ˚.ˇFˇ/ ; (14.14) ˇ ˇ where ˇ0 and ˇ are the constants in .14.1/ : Moreover, ˇV ; F V.0; F/ˇ Ä 2ˇ j 0j .1 C ˚.jFj//: Moreover, the function QVin(14.8) is continuous and satisﬁes (14.14), and

jQV.; F/ QV.0; F/jÄCj0 j.1 C ˚.jFj//: (14.15)

A crucial tool in the proof of Theorem 1 is the following Decomposition Lemma (see [KoEtAl17]). Lemma 2 (Scaled Decomposition Lemma) Let ! 2 be a bounded open set with Lipschitz boundary and ˝ WD ! .1; 1/. Let ˚ W Œ0; C1/ ! Œ0; C1/ be 1;˚ 3 an Orlicz function satisfying (14.9) and (14.10). Let .u"/ W .˝I /. Assume that ."/ C is a sequence of numbers converging to 0, such that Z ˇ ˇ .˚.ˇ ; 1 ˇ/ < : sup r˛u" " r3u" dx D C C1 " ˝

Then there exist a (non-relabelled) subsequence .u"/ and a sequence .v"/ W1;˚ .˝I 3/ such that ˇ ˇÁ ˚ ˇ v ; 1 v ˇ (i) sequence r˛ " " r3 " is equi-integrable, 1;˚ 3 1;˚ 3 (ii) v" * u0 in W .˝I /; where u0 is the weak limit of .u"/ in W .˝I /; (iii) jfx 2 ˝ W u" ¤ v" or ru" ¤rv"gj ! 0; as " ! 0, (iv) v"j@!.1;1/ D u0: 14 Optimal Design in Orlicz–Sobolev Setting 167

14.3 Proof of Theorem 1

We start by motivating the choice of the topology in Theorem 1. We claim that 1;˚ 3 energy bounded sequences ."; u"/ 2 BV.˝If0; 1g/ W .˝I /, admissible for the rescaled version of (14.2), i.e. such that there exists C >0W ˇ Z ˇ ˇ ˇ ˇ ˇ Œ. .1 / / ; 1 ˇ ; 1 ˇ .˝/ ; ˇ "W1 C " W2 r˛u" " r3u" C f u" dx C D˛ " " D3 " ˇ Ä C ˝ R (14.16) @! . 1; 1/ 1 with u" clamped on and j˝j ˝ "dx D , are compact in space L1.˝If0; 1g/ L˚ .˝I 3/ and with limit in BV.!If0; 1g/ W1;˚ .!I 3/. Indeed, 0 C let ."; u"/ be a sequence such that (14.16) holds, then there exists C 2 such that ˇÂ Ãˇ 1 ˇ 1 ˇ 0 0 ˇ ˇ 0 u" 1;˚ C ; 3u" C ; D˛"; D3" .˝/ C : k kW Ä " r Ä ˇ " ˇ Ä (14.17) L˚

Then, standard arguments in dimensional reduction (see [LeEtAl95, Lemma 3]) 1;˚ 3 entail that there exists u 2 W .˝I / such that r3u Á 0, and so u can be identiﬁed with a function (still denoted in the same way, cf. [Ma11, Theorem 1 in Section 1.1.3]) u 2 W1;˚ .!I 3/: Analogous considerations hold for the limit of ". Thus there exists a subsequence, not relabelled, ."; u"/ such that 1;˚ 3 ? u" * u in W .˝I /, and a measurable set E ˝ such that " *E and 0 D3E Á 0: Hence, there exists E !, with jDEj.˝/ D 2jDE0 j.!/; where E D E0 .1; 1/: In the sequel we will identify the set E with the set E0 and denote E0 by . We observe that Theorem 1 still holds without the coercivity assumption (14.1), provided the admissible sequences satisfy (14.17).

Proof (Theorem 1) For every ">0,letJ" be the functional in (14.4). The separability of the metric space L1.˝If0; 1g/ L˚ .˝I 3/ ensures that for each sequence ."/ there exists a subsequence, still denoted by ."/, such that . 1.˝ 0; 1 / ˚ .˝ 3// lim"!0C L If g L I J" exists. For every .; u/ 2 L1.˝If0; 1g/ L˚ .˝I 3/,letJ .; u/ be this -limit. By Urysohn property, it sufﬁces to prove that any sequence .J"/ admits a further subsequence whose -limit, J.; u/; coincides with J0.; u/ in (14.5). It is easily seen that if .; u/ 2 .L1.˝If0; 1g/ L˚ .˝I 3// n .BV.!I f0; 1g/ W1;˚ .!I 3//, then J.; u/ DC1. Indeed, if this is not the case, from J.; u/< C1 we would get the existence of a sequence ."; u"/ converging to .; u/ such 1;˚ 3 that J"."; u"/

1 ˚ 3 To prove the claim, let ."; u"/ L .˝If0; 1g/ L .˝I / be a sequence converging to .; u/ 2 BV.!If0; 1g/ W1;˚ .!I 3/. For the forces and for the ˚ perimeter the lower bound follows by L strong convergence of .u"/, the continuity of the linear term, and lower semicontinuity of the total variation, respectively. For what concerns the bulk energy, by Lemma 2, there exist .w"/ and .A"/, such ˚ 3 1;˚ 3 that A" ˝, w" converges in L .˝I / to u 2 W .!I /, the scaled gradients . ; 1 / ˚ ˝ ˝ 0 r˛w" " r3w" are -equi-integrable, A" Â and u" Á w" in A" and j n A"j! C as " ! 0 . Denoting the bulk energy density of J" by V as in (14.6), one obtains R ; 1 lim inf"!0C R˝ V " r˛u" " r3u" dx ; ; 1 lim inf"!0C ˝ VR " r˛w" "ˇr3w" dx ˇ ˇ 1 ˇ ˇ lim sup C .1 C ˚. .r˛w"; r3w"/ //dx (14.18) "!R 0 ˝nA" " ; ; 1 lim inf"!0C R˝ V " r˛w" " r3w" dx R . ; / . ; / ; lim inf"!0C ˝ V " r˛w" dx lim inf"!0C ˝ QV " r˛w" dx where the latter density is the one deﬁned in (14.8) Observethat,by(14.15), thus it is quasiconvex in 32 for a.e. x 2 ˝. Z Z

jQV ."; r˛w"/ QV .; r˛w"/ j dx Ä C j" j.1 C ˚.jr˛w"j// dx: ˝ ˝ ˇ ˚ . ˇ 1 / " 0C Thus, the -equi-integrability of r˛w" " r3w" ensures that as ! , " can be replaced by in the right-hand side of (14.18). From [Go82, Theorem 1] it follows that smooth functions are dense in W1;˚ .E/ providing ˚ satisﬁes (14.10) and the argument exploited in [LeEtAl95, Propo- 33 sition 6] ensures that QV..x˛/; / is quasiconvex also in . Thus, by the growth condition of QV;Rin (14.14), and [Fo97, Theorem 3.1] the functional 1;˚ 3 v 2 W .˝I / 7! ˝ QV..x˛/; r˛v.x//dx is sequentially weakly lower semicontinuous with respect to W1;˚ -weak topology (and, by (14.13), strongly in L˚ ). Hence, Z Z

lim inf QV.; r˛w"/dx 2 QV.; r˛u/dx˛: "!0C ˝ !

By the superadditivity of the lim inf we achieve the claim. Upper bound: To prove that J.; u/ Ä J0.; u/ for every .; u/ 2 BV.!If0; 1g/ W1;˚ .!I 3/; we start by observing that for every 2 BV.!If0; 1g/, J.; u/ Ä .; / ˚ .˝ 3/: lim inf"!0C J" u" for every u" ! u in L I Thus we can reduce to study the asymptotic behaviour with respect to the W1;˚ - weak convergence of Z Z ; 1 .1 / ; 1 : W1 r˛u" " r3u" C W2 r˛u" " r3u" dx f u"dx (14.19) ˝ ˝

Since is ﬁxed we can rewrite W1./C.1/W2./ as a new function with explicit dependence on x˛. 14 Optimal Design in Orlicz–Sobolev Setting 169

Denoting V..x˛/; F/ by W.x˛; F/, it results that it is a Carathéodory function 1 ˚. / . ; / satisfying a growth condition of the type (14.1), i.e. C jFj C Ä W x˛ F Ä C 33 C.1 C ˚.jFj// for a suitable constant C 2 , for a.e. x˛ 2 ! and for all F 2 : Next we can argue as in [BrEtAl00, Theorem 2.5] and [BaEtAl05, Lemma 2.5]. ˚ 3 We deﬁne the sequence of functionals .G"/, where G" W L .˝I / ! Œ0; C1/ is given by R R ; ; 1 1;˚ .˝ 3/; ˝ W x˛ r˛u " r3u dx ˝ f udx if u 2 W I G".u/ D C1 otherwise, and we claim that ÂZ Z Ã ; ; 1 lim W x˛ r˛u" " r3u" dx f u"dx Ä "!0C ˝ ˝ Z Z 1 Z W.x˛; r˛u/dx˛ f udx˛dx3; ! 1 ! where W W ! 32 is deﬁned by

W.x˛; F/ n R WD 1 inf W.x˛; F Cr˛'.y˛; y3/; r3'.y˛; y3//dy˛dy3 W 2 Q.1;1/ o (14.20) 1;˚ 0 0 ' 2 W0 .Q .1; 1//; ' D 0 on @Q .1; 1/; > 0 :

˚ 3 By the strong convergence of u" ! u in L .˝I / it suffices to focus on the bulk term. To this end we observe that there exists a subsequence of .G"/ which -converges to a functional G which is a measure. Indeed with straightforward modifications to the argument of [BrEtAl00,Step3. and Step 4. Lemma 2.6] (which can be adapted to W1;˚ setting) one can prove that for every u 2 W1;˚ .!; 3/, the set function G.u; / W A.!/ ! , defined on open subsets A Â !, satisfies (i) G.u; A/ D G.vI A/ whenever u D v; a.e. on 2, (ii) G.u; / is the restrictionR of a finite non-negative Radon measure on A.!/, G.u; A/ 2ˇ .1 ˚. D˛u //dx (iii) Ä A C j R jR , . / . ; / 1 3 (iv) G u C cI A D G u A C 1 A f cdx˛ for any c 2 . This is enough to apply the integral representation theorem by Buttazzo and Dal Maso [Bu89, Theorem 4.3.2]. Indeed this latter result holds without any substantial modifications in our Sobolev-Orlicz setting, in particular all the crucial steps, (as ‘Zig-Zag’ lemma or the passage through affine and piecewise affine functions) can be repeated word by word. In particular we emphasize that the approximation of functions in Sobolev-Orlicz spaces (with Orlicz function ˚ 2 2), by piecewise ones holds as originally stated in [EkEtAl76, Proposition 2.8]. 170 P.A. Kozarzewski and E. Zappale

e ! 32 . ; / R The existence ofR an energyR density W W ! such that G u A D e. ; / 1 A W x r˛u dx˛ 1 A f udx˛ is thus guaranteed. Finally the same type of estimates as in [BaEtAl05, Lemma 2.5], up to replacing Sobolev spaces by Sobolev- e 32 Orlicz ones, entails that W.x˛; F/ Ä W.x˛; F/, for a.e. x˛ 2 ! and every F 2 , where W is the density in (14.20). 32 . ; / . ;. //; Next, introduce for every F 2 , W x˛ F WD infc23 W x˛ Fjc and denote by QW the quasiconvexiﬁcation of W with respect to the second variable, according to (14.8). Finally, arguing as in Remark 3.3 of [BrEtAl00, Remark 3.3] for a.e. x˛ 2 ! 32 and every F 2 we get W.x˛; F/ D QW.x˛; F/, (we refer to the proof of [KoEtAl17, Theorem 2] for more details). The proof is concluded observing that by (14.7)

W.x˛; F/ D .x˛/W1.F/ C .1 .x˛//W2.F/ D V..x˛/; F/

32 and QW.x˛; F/ D QV..x˛/; F/ for every .x˛; F/ 2 ! .

Acknowledgements P.A.K.’s research was supported by WCMCS, http://www.wcmcs.edu.pl/ and by INdAM-GNAMPA through the project ‘Professori Visitatori 2016’. The support and the hospitality of University of Salerno and University of Warsaw is gratefully acknowledged by both authors. Elvira Zappale is a member of INdAM-GNAMPA.

References

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C.A. Ladeia, J.C.L. Fernandes, B.E.J. Bodmann, and M.T. Vilhena

15.1 Introduction

The equation of radiative conductive transfer in cylinder geometry has numerous engineering applications, including the radiation transfer in furnaces, chimney stacks, rocket engines, stellar atmospheres and other similar situations [Mo83]. Solutions found in the literature are typically determined by numerical means, see, for instance, refs. [LiOz91, Li00, MiEtAl11]. Recently, the radiative conductive transfer equation in cylinder geometry was solved in semi-analytical fashion by the collocation method in both angular variables, using the SN procedure [LaEtAl15]. The solution was constructed using Laplace transform together with a decomposition method [ViEtAl11]. The Laplace method is related to procedures for linear problems, while the decomposition method allows to treat the non-linear contribution as source term in a linear recursive scheme and thus opens a pathway to determine a solution, in principle to any prescribed precision [PaEtAl03, ViEtAl11]. In the present discussion, we demonstrate heuristic convergence of the solution obtained by a recursive scheme, inspired by a stability analysis but for convergence.

C.A. Ladeia () • J.C.L. Fernandes • B.E.J. Bodmann • M.T. Vilhena Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil e-mail: [email protected]; [email protected]; [email protected]; [email protected]

15.2 The Radiative Conductive Transfer Equation in Cylinder Geometry

We consider the one-dimensional steady state problem in cylindrical geometry for a solid cylinder. The problem of energy transfer is described in [Oz73]by the radiative conductive transfer equation coupled with the energy equation, with T D .1 !.r// 4.r/, Ä Â Ã p @ 1 2 @ 1 2 C C 1 I.r;; /D @r r @ Z Z !.r/ 1 1 d 0 D P.; 0/I.r;0; 0/d0 p C T ; (15.1) 2 1 0 1 0 for r 2 .0; R/, 2 Œ0; 1 and 2 .1; 1/. Here, D cos./, is the polar angle, D cos. /, is the azimuthal angle, I is the radiation intensity, ! is the single scattering albedo and P.; 0/ signiﬁes the differential scattering coefﬁcient, also called the phase function [Ch50]. In this work, we consider polar symmetry due to translational symmetry along the cylinder axis. Moreover, the isotropic medium implies azimuthal symmetry. The integral on the right-hand side of (15.1) can be written as

Z 1 XL Z 1 0 0 0 0 0 0 0 0 P.; /I.r; ; /d D ˇl Pl./Pl. /I.r; ; /d ; 0 0 lD0 where ˇl are the expansion coefﬁcients of the Legendre polynomials Pl./ and l refers to the degree of anisotropy, for details see [ViEtAl11]. The energy equation for the temperature that connects the radiative ﬂux to a temperature gradient is

d2 d 1 d . / . / Œ : r 2 r C r D rqr rH (15.2) dr dr 4 Nc dr

kˇext Here, Nc D 4n2 3 is the radiation conduction parameter with k the thermal Tr conductivity, ˇext the extinction coefﬁcient, the Stefan-Boltzmann constant, n the 1 refractive index, Tr is a reference temperature, H D ŒkˇextTr h is the normalised constant, and h is used to denote prescribed heat generation in the medium that is independent of the radiation intensity. The dimensionless radiative heat ﬂux is expressed in terms of the radiative intensity by Z Z 1 1 d 0 q.r/ D 4 I.r;0; 0/d0 p : r 2 0 1 1 0 15 A Heuristic Convergence Criterion by Stability Analysis 175

The boundary conditions of equation (15.1)are

I.0;; /D I.0; ; / ; Z Z 4d.R/ 1 1 p d 0 I.R;; /D .R/4.R/ C I.R;0; 0/ 1 02d0 0 p ; 0 0 1 02 for 2 Œ0; 1 and 2 Œ0; 1/, d is the diffuse reﬂectivity, is the emissivity, the thermal photon emission according to the Stefan-Boltzmann law (see ref. [El09]), and the boundary conditions of equation (15.2)are ˇ ˇ d . /ˇ 0 . / : r ˇ D and r jrDR D B (15.3) dr rD0

In addition, if the radiative heat ﬂux is known, we can solve (15.2) and use equations (15.3) to ﬁnd Z H 1 R . / 2 2 . 0/ 0 : r D B C R r qr r dr (15.4) 4 4 Nc r

15.3 Solution by the Decomposition Method

In order to obtain a solution, we use the SN approximation extended by an additional angular variable In;m Á In;m.r;n; m/.TheSN approximation [Ch50] is based on the angular variable discretisation ˝ in an enumerable set of angles or equivalently their direction cosines, in our work n and m. Then, equations (15.1) and (15.2) can be simpliﬁed using an enumerable set of angles following the collocation method, which deﬁnes the problem of radiative conductive transfer in cylindrical geometry in the so-called SN approximation, Â Ã ˇ @ 1 2 @ ˇ 1 In;m m I ˇ m C ˇ C p In;m D (15.5) @r r @ 1 2 Dn; D m n XL XN=2 XN !.r/ .1 !.r// 4 D p ˇ P . / $ ; P . /I ; C p .r/; 1 2 l l n p q l p p q 1 2 n lD0 pD1 qD1 n

ˇ ˇ XN=2 XN d d ˇ 1 .r/ .r/ˇ D $p;qŒIp;q.r/ Ip;q.0/ : dr dr 0 N rD c pD1 qD1

Here, n indicates a discrete direction of n and m a discrete direction of m, respectively. The integration is carried out over two octants with 1 Ä n Ä N=2 176 C.A. Ladeia et al. and 1 Ä m Ä N. In this approximation equation (15.5) may be written as Â Ã ˇ @ 1 2 @ ˇ 1 In;m m I ˇ m C ˇ C p In;m D .r;n; n;m/; (15.6) @r r @ 1 2 Dn; D m n XL XN=2 XN !.r/ .1 !.r// 4 .r; ; / D p ˇ P . / $ ; P . /I ; C p .r/: n m 1 2 l l n p q l p p q 1 2 n lD0 pD1 qD1 n Á ˇ @I 1 2 ˇ n;m m @I Instead of m C ˇ , we take as independent variables X D @r r @ ; p D n D m Y 1 2 1 =2 1 r m and D r m,for Ä n Ä N and Ä m Ä N. Then equation (15.6) @In;m 1 becomes C p I ; D .X; Y; / or in S representation @X 1 2 n m n N n

1 @I ; I ; n m C p n m D @ 1 2 m r n XL XN=2 XN !.r/ .1 !.r// 4 p ˇ P . / $ ; P . /I ; C p .r/; 1 2 l l n p q l p p q 1 2 n lD0 pD1 qD1 n

ˇ ˇ XN=2 XN d d ˇ 1 .r/ .r/ˇ D $p;qŒIp;q.r/ Ip;q.0/ : (15.7) dr dr 0 N rD c pD1 qD1

Here, n and m are evaluation points, with 1 Ä n Ä N=2 and 1 Ä m Ä N and subject to the following boundary conditions:

In;m.0/ DIp;NmC1.0/ ; =2 =2 4d. / XN XN q 4 R 2 I ; 1.R/ .R/ .R/ $ ; I ; .R/ 1 q: n NmC D C p q p q p pD1 qD1

Note that the integrals over the angular variables are replaced by Gauss-Legendre $ $ p and Gauss-Chebyshev quadratures with weight p;q D N , with Gauss-Legendre P =2 weight normalisation N $ D 1 and Gauss-Chebyshev weight normalisation to pD1 Pp P N=2 N $ the solid angle of an octant pD1 qD1 p;q D . The choice for the speciﬁc quadratures is due to the phase function representation in terms of Legendre polynomials, and the second quadrature scheme by the singular structure of the integral, that is found in representations for Chebyshev polynomials. The equation system (15.7) may be cast in a ﬁrst order matrix equation

d A I BI D ; (15.8) dr 15 A Heuristic Convergence Criterion by Stability Analysis 177

N2 N2 1= where A is a diagonal matrix of order 2 2 , with A D n;m, B is a square matrix of the same order as A with elements ! p XL Bi;j D 1= 1 i ıij C !.r/ ˇlPl.i/Pl.j/ ; lD0

ı N2 ij is the usual Kronecker symbol andÁ the vector of the intensity of order 2 is T deﬁned by I D I1;1; ; I1; ; ; I N . The non-linear terms are N sequences of N 2 ;N N identical angular terms for the N=2 directions. 0 1 T B.1 !.r// .1 !.r// C D @ q 4.r/;:::; q 4.r/A : 1 2 1 2 1 N=2

AccordingP to [ViEtAl11] the intensity of radiation is expanded in an infinite 1 series I D lD0 Yl, which introduces an infinite number of artificial degrees of freedom and may be used to set up a non-unique recursive scheme of linear differential equations, where the non-linear terms appear as source terms containing known solutions Yl from the previous recursion steps [ViEtAl11] and the solution of the linear differential equations is known. Equation (15.8) is then 0 1 Â Ã T X1 d B.1 !.r// .1 !.r// C X1 q ;:::;q l1 : A YlBYl D @ A Gl1 fYlglD0 2 2 0 dr 1 1 0 lD 1 N=2 „lDƒ‚… 4.r/ (15.9)

In order to solve the equation system (15.9) in a recursive fashion, the initialisation d 0 is chosen to be A dr Y0 BY0 D further subject to the original boundary conditions and then making use of a recursive process of the equations for the remaining components Yl, with homogeneous boundary conditions. 0 1 T d B.1 !.r// .1 !.r// C q ;:::; q l1 ; A Yl BYl D@ A Gl1 fYlglD0 (15.10) dr 1 2 1 2 1 N=2 with l 2 C. Accordingly, the recursion initialisation corresponds to the homogeneous solution, where all subsequent recursion steps result in particular solutions. The remaining, L1 denotes the inverse Laplace transform operator, s is the r dual complex variable from Laplace transform of equation (15.10), U D A1B and the decomposed matrix U D XDX1 with D the diagonal matrix with 178 C.A. Ladeia et al. distinct eigenvalues and X the eigenvector matrix. The general solution for each decomposition term Yl.r/ is explicitly given by

1 1 1 Yl.r/ D L ..sI U/Yl.0// C L ..sI U/A .s// 0 1 T .1 !. // .1 !. // Dr Dr 1 1 B r r C D Xe Yl.0/ C Xe X A @ q ;:::; q A Gl1: 1 2 1 2 1 N=2 P 4. / J l1 The non-linearity from the temperature term r D lD0 Gl fYlglD0 is l1 represented by Adomian polynomials Gl fYlglD0 . In the following the notation .l/ f0 stands for the l-th functional derivative of the non-linearity at Y D Y0, so that .0/ one may identify the first term f0 and all the subsequent terms of the series that define the Adomian polynomials Gl as shown below (for details see [ViEtAl11]). Â Z Ã R 4 .0/ H 1 . / . / . 2 2/ . 0/ 0 ; G0 r D f0 D f Y0 D B C R r q0;r r dr 4 4 Nc r Â Z Ã R 4 .1/ d 1 . / . / . 0/ . 0/ 0 G1 r D f0 Y1 D Y1 f Y0 D q0;r r C q1;r r dr dY0 4 Nc r 2 .1/ .2/ 2 d Y1 d G2.r/ D f0 Y2 C f0 Y1 D Y2 f .Y0/ C f .Y0/ D dY0 2Š dY0 Â Z Ã 1 R 4 . 0/ . 0/ . 0/ 0 D q0;r r C q1;r r C q2;r r dr 4 Nc r : : ! Xl X Yl1 .1/ 1 . / j . / j b : Gl r D f0 Yl C f0 l1 Y (15.11) jŠ fbig1 jD2 b1P;:::;bl1 D1 biDj ! j Here l1 are the usual multinomial coefficients. In principle one has to solve fbig1 an infinite number of equations, so that in a computational implementation one has to truncate the scheme according to a prescribed precision. In the present state of the work the pertinent question of convergence in Section 15.4 is based on heuristic arguments. From the last term in equation (15.11) one observes that for a non- linearity with polynomial structure there is only a limited number of combinations in the last term. Consequently, for j sufficiently large the factorial term jŠ controls the magnitude of the correction terms, that for increasing j tend to zero. One may now determine the temperature profile. To this end and in order to stabilise convergence 15 A Heuristic Convergence Criterion by Stability Analysis 179 a correction parameter ˛ for the recursive scheme was introduced, where 1 D Z˛, Nc with Z 2 , Z X1 H Z˛ X1 R .r/ .R2 r2/ q .r0/dr0 : l D B C 4 4 l;r lD0 lD0 r

15.4 A Heuristic Convergence Criterion by Stability Analysis

In general convergence is not guaranteed by the decomposition method, so that the solution shall be tested by a convenient criterion. Since standard convergence criteria do not apply for non-linear problems, we translate the Lyapunov-Boichenko stability criterion [BoEtAl05] for dynamical systems into a stability criterion for convergence of recursive schemes. To this end let X1 jıZJ j D l ; lDJC1 represent the majorP difference between the solution obtained with truncation J,in J this case, J D lD0 l, and the true solution, with kkthe maximum norm. Therefore

J jıZJ j D e jıZ0j ; shows how the difference of the recursion initialisation to the true solution evolves until recursion depth J. If the exponent remains negative <0for increasing J the series is convergent Â Ã 1 ıZ j J j : D P ln J jıZ0j lD0 l Â Ã P ıZ J j J j Here lD0 l is known and ln jıZ0j is estimated. The general term l is expressed by Ä Z ˛ XM R H 2 2 0 0 .r/ ı0; .R r / q .r /dr ; l D l B C 4 4 li;r (15.12) iD1 r where ı0;l is the Kronecker symbol and M D min.Z; l/. The last term in this equation (15.12) contains all the recurrences of i for i 1. 180 C.A. Ladeia et al.

15.5 Numerical Results

To check if the proposed method is appropriate for radiative conductive transfer problems in a solid cylinder, we evaluate the normalised temperature, conductive, radiative and total heat ﬂuxes 1 r 1 r . / ; . / ; . / : Qr r D qr Qc r D H qr Q r D H 4 Nc 2 4 Nc 2

All the numerical results are based on the parameter set with ! D 0:9, D 0:9, d D 0:1, B D 1, R D 0:5 and r in units of r=R that varies between 0 and 1. The numerical results for the profile of the temperature ./,forthe conductive heat flux .Qc/, the radiative heat flux .Qr/, the total heat flux .Q/ along the radial optical depth, and also the depth of the finite recursion and the stability of finite recursion are shown in Figures 15.1, 15.2 and 15.3, respectively, where we consider .Nc; H;˛/ Df.0:05; 0; 10/; .0:005; 40; 1/; .0:05; 100; 0:1/g. In this analysis N D 4 directions were used. Figure 15.1-(left) shows that the influence of an increase in the temperature is associated with the parameters Nc and H. Note that the conductive heat flux, Figure 15.1-(right), passes through the minimum point and already the radiative heat flux, Figure 15.2-(left) has a corresponding maximum point and the total heat flux, Figure 15.2-(right) (see above). Moreover, Figure 15.3-(left) shows the temperature after recursion steps for the set .Nc; H;˛/ Df.0:05; 0; 10/; .0:005; 40; 1/; .0:05; 100; 0:1/g.Weuseda stopping criterion such that the last twenty recursions of temperature change less than 106. Accordingly the series of Adomian polynomials is truncated at J D 45, J D 9557, and J D 6361 for each element. Figure 15.3-(right) shows the stability of the proposed methodology, where the Lyapunov criterion has been met for truncation orders J D 45, J D 9557 and J D 6361. Furthermore, although the recursive scheme shows an oscillatory character, the solution is already stable before truncation, given that is negative.

3 12 α Nc=0.05; H=0; =10 10 α 2.5 Nc=0.005; H=40; =1 α α 8 Nc=0.05; H=100; =0.1 Nc=0.05; H=0; =10 2 N =0.005; H=40; α=1

Θ c 6 N =0.05; H=100; α=0.1 c c Q 1.5 4 2 1 0 0.5 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r/R r/R

Fig. 15.1 The temperature proﬁle (left) and the conductive heat ﬂux Qc (right) against the relative radial optical depth 15 A Heuristic Convergence Criterion by Stability Analysis 181

25 16 N =0.05; H=0; α=10 N =0.05; H=0; α=10 c 14 c 20 N =0.005; H=40; α=1 N =0.005; H=40; α=1 c c N =0.05; H=100; α=0.1 12 α c N =0.05; H=100; =0.1 15 10 c 8 r 10 Q Q 6 4 5 2 0 0 -2 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r/R r/R

Fig. 15.2 The radiative heat ﬂux Qr (left) and the total heat ﬂux Q (right) against the relative radial optical depth

8 14 α Nc=0.05; H=0; =10 N =0.05; H=0; α=10 7 α 12 c Nc=0.005; H=40; =1 N =0.005; H=40; α=1 6 α c Nc=0.05; H=100; =0.1 10 α Nc=0.05; H=100; =0.1 5 8 λ

4 - 6 3 4 2 2 Recursive temperature 1 0 0 -2 0 1 2 3 4 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 Recursion depth Recursion depth

Fig. 15.3 Recursive temperature at r=R for recursion depth (left) and negative exponent dependence on recursion depth (right)

15.6 Conclusions

The present study presents a heuristic convergence for the radiative conductive transfer problem in cylinder geometry, that was cast into an approximate problem with discretised angular variables using the SN procedure. The original non-linear problem was decomposed in a recursive scheme of equation systems. To this end a modiﬁed decomposition procedure was used, following the reasoning of reference [ViEtAl11]. The initialisation of the recursion is a linear equation system with known solution. All the subsequent equation systems to be solved are of linear type, where the non-linearity appears as source term but containing only terms with the solutions from all previous recursion steps. It is noteworthy that the recursive scheme is not unique and convergence depends strongly on the recursion initialisation. In the present work the heuristic convergence is implemented in analogy to Lyapunov’s idea for dynamical systems. Convergence may be indifferent chaotic or stable depending on the sign of >0, D 0, <0. Recursion truncation applies if 0 for all subsequent recursion steps. From the generic decomposition scheme it seems plausible that convergence occurs for problems with non-linearities of polynomial type. In the considered cases the speciﬁc parameter choice needs a corrective procedure for the recursion initialisation. The present approach has the merit that it works for general cases, which is not the case for “so-called” benchmark calculations from the literature. 182 C.A. Ladeia et al.

References

[BoEtAl05] Boichenko, V.A., Leonov, G.A., Reitmann, V.: Dimension Theory for Ordinary Differential. Teubner, Stuttgart (2005) [Ch50] Chandrasekhar, S.: Radiative Transfer. Oxford University Press, New York (1950) [El09] Elghazaly, A.: Conductive-radiative heat transfer in a scattering medium with angle- dependent reﬂective boundaries. J. Nucl. Radiat. Phys. 4, 31–41 (2009) [LaEtAl15] Ladeia, C.A., Bodmann, B.E.J., Vilhena, M.T.B.: The Radiative-Conductive Transfer Equation in Cylinder Geometry and Its Application to Rocket Launch Exhaust Phenomena. Integral Methods in Science and Engineering. Springer, Cham (2015). ISBN: 978-3-319-16727-5. doi:10.1007/978-3-319-16727-5_29 [Li00] Li, H.-Y.: A two-dimensional cylindrical inverse source problem in radiative transfer. J. Quant. Spectrosc. Radiat. Transf. 69, 403–414 (2000) [LiOz91] Li, H.Y., Ozisik, M.N.: Simultaneous conduction and radiation in a two-dimensional participating cylinder with anisotropic scattering. J. Quant. Spectrosc. Radiat. Transf. 46, 393–404 (1991) [MiEtAl11] Mishra, S. C., Krishna, Ch. H., and Kim, M. Y.: Analysis of conduction and radiation heat transfer in a 2-D cylindrical medium using the modiﬁed discrete ordinate method and the lattice Bolztmann method. Numer. Heat Transf. 60, 254–287 (2011) [Mo83] Modest, M.F.: Radiative Heat Transfer. McGraw-Hill, New York (1993) [Oz73] Ozisik, M.N.: Radiative Transfer and Interaction with Conductions and Convection. Wiley, New York (1973) [PaEtAl03] Pazos, R.P., Vilhena, M.T., Hauser E.B.: Advances in the solution of threedimensional nodal neutron transport equation. In: 11th International Conference of Nuclear Engineering, Tokyo (2003) [ViEtAl11] Vilhena, M.T.M.B., Bodmann, B.E.J., Segatto, C.F.: Non-linear radiative-conductive heat transfer in a heterogeneous gray plane-parallel participating medium. In: Ahasan, A. (ed.) Convection and Conduction Heat Transfer. InTech, New York (2011). ISBN: 978-953-307-582-2. doi:10.5772/22736 Chapter 16 An Indirect Boundary Integral Equation Method for Boundary Value Problems in Elastostatics

A. Malaspina

In this paper, following [Ci88], we apply an indirect boundary integral equations method to solve the Dirichlet problem for the n-dimensional linearized elastostatic system in a multiply connected domain of n, n 2. In particular we show how to represent the solution in terms of a single-layer potential, instead of the usual double-layer potential. As a consequence, we are able to treat also the double- layer potential ansatz for the traction problem. We point out that such method only requires the theory of reducible operators (see, e.g., [MiPr86]) as well as the theory of differential forms (see, e.g., [Fi61, Fl63]).

16.1 Preliminaries

In this section, we introduce the quantities that will be the focus of our study regarding the linearized n-dimensional elasticity theory. For further details on these arguments and associated potential theory see, e.g., [KuEtAl79, HsWe08]. Let us consider ˝ a domain (open connected set) of n, n 2. The displacement vector u W ˝ ! n satisﬁes the system of equations for the linearized elastostatics if and only if

Eu.x/ D u.x/ C krdivu.x/ D 0; x 2 ˝; (16.1)

A. Malaspina () University of Basilicata, Potenza, Italy e-mail: [email protected]

P @2 >. 2/= n ui . @ ;:::; @ / where k n n is a real constant, ui D hD1 @ 2 , rD @x1 @x and P xh n n @uh div u D 1 : Moreover, we denote the boundary of ˝ with the symbol ˙ hD @xh and we use W ˙ ! n to represent the outward unit normal vector. Given a real parameter , we introduce the boundary operator L W ˙ ! n whose components are . /. / @ @ ; 1;:::; ; Li u D k div u i C j jui C j iuj i D n (16.2) where ıij is the standard Kronecker symbol. L can be considered as the generalized stress operator of elasticity, in fact, for D 1 the operator L1 is just the classical stress, that we shall simply denote by L,while,for D k=.k C 2/, Lk=.kC2/ is the pseudo-stress operator. As a matrix of fundamental solutions for .E/ we consider the Kelvin’s matrix with entries 8 Â Ã . /. / ˆ 1 k C 2 k xi yi xj yj <ˆ ıij ln jx yjC ; if n D 2; 2 2.k C 1/ 2.k C 1/ jx yj2 ! .x; y/ 2 ij D ˆ 1 k C 2 jx yj n k .x y /.x y / ˆ ı i i j j ; 3; : ij C n if n !n 2.k C 1/ 2 n 2.k C 1/ jx yj (16.3) (x ¤ y), i; j D 1;:::;n, !n being the hypersurface measure of the unit sphere in n. Now we introduce the classic single- and double-layer potentials which are fundamental in potential theory for elasticity. The elastic single-layer potential of density ' W ˙ ! n is deﬁned by Z

VŒ'.x/ D .x; y/'.y/ dy; x … ˙ (16.4) ˙ and the elastic double-layer potential with density W ˙ ! n is given by Z 0 WŒ .x/ D .Ly.x; y// .y/ dy; x … ˙; (16.5) ˙

0 where ŒLy.x; y/ denotes the transposed matrix of LyŒ .x; y/ D @yŒ .x; y/. The basic boundary value problems (BVPs) related to elastostatic system (16.1) are two. The ﬁrst one is the Dirichlet problem where the displacement u is prescribed on the boundary: Eu D 0 in ˝; (16.6) u D f on ˙:

The second one is the traction problem where the stress is equal to an assigned vector function f on the boundary: Eu D 0 in ˝; (16.7) Lu D f on ˙; 16 An Indirect Boundary Integral Equation Method in Elastostatics 185 where f satisﬁes the following condition Z f ˛ d D 0 (16.8) ˙ ˛ being an arbitrary rigid displacement.

16.2 Indirect Method

The classical indirect method for solving the two basic BVPs (16.6) and (16.7) consists in seeking the solution in the form of an elastic double-layer potential (16.5) for the Dirichlet problem and in the form of an elastic single-layer potential (16.4) for the traction problem. It is well known that, if the boundary is sufficiently smooth, in both cases it leads to a singular integral system which can be reduced to a Fredholm one (see, e.g., [KuEtAl79]). Here we illustrate an indirect boundary integral equations method which consists in looking for the solutions of the two BVPs in integral forms different from the usual ones. In particular we study the possibility to represent the solution of (16.6) by means of an elastic single-layer potential. This approach leads to an integral system of equations of the first kind on the boundary which can be treated in different ways. Here we apply an indirect method (introduced in [Ci88] for Dirichlet problem of Laplace equation) in which neither the knowledge of pseudo-differential operators nor the use of hypersingular integrals is required, it hinges only on the theory of differential forms and on the theory of reducible operators. From now onS we assume that ˝ is a multiply connected domain, that is, ˝ has the ˝ ˝ m ˝ ; ˝ 0;:::; 1 n form D 0n jD1 j where j (j D m)aremC bounded domains of , 2 ˝ ˝ ˝ ˝ ; 1;:::; ; n , such that j 0 and Sj \ k D;, j k D m j ¤ k. Moreover, we ˙ m ˙ ˙ @˝ suppose that the boundary D jD0 j is a Lyapunov hypersurface ( j D j 2 C1;, 2 .0; 1). We note that the usual integral representations of solutions to classical BVPs for some partial differential systems in multiply connected domains have been object of several papers in the last decades (see, e.g., [MaEtAl99, StTa02, Ko05, Ko07]). p.˙/ Before describing the method, we introduce the symbol Lh (for any nonnegative integer h) which denotes the vector space of all differential forms of degree h defined on ˙ such that their components are integrable functions belonging to Lp.˙/ in a coordinate system of class C1 (and consequently in every coordinate system of class C1). Dirichlet Problem. We consider the Dirichlet problem (16.6) with data f in the Sobolev space ŒW1;p.˙/n (1

.x; y/'.y/dy D f .x/: (16.9) ˙ 186 A. Malaspina

Following [Ci88], if we take the differential of both sides of (16.9), then a singular integral system comes out: Z

dxŒ .x; y/'.y/ dy D df .x/; (16.10) ˙ where dx stands for the exterior differentiation. The peculiarity of this system lies in the fact that the unknown is a vector function ' 2 ŒLp.˙/n, while the data df is a vector belonging to an other space: df is a vector whose components are differential form of degree 1 with coefﬁcients in Lp space. We show that the singular integral system (16.10) can be reduced to a Fredholm one. In fact, set the singular integral p n p n operator R W ŒL .˙/ ! ŒL1.˙/ as Z

R.'/.x/ D dxŒ .x; y/'.y/ dy: (16.11) ˙ Then, it can be reduced on the left, i.e., there exists a linear and continuous operator p n p n p n RQ W ŒL1.˙/ ! ŒL .˙/ such that RRQ is a Fredholm operator from ŒL .˙/ into itself. In order to determine a reducing operator for R, let us introduce the singular p n p n integral operator R W ŒL1.˙/ ! ŒL .˙/ , 2 , whose components are:

. /. / . /K . /. / . / . /K . /. / . /K . /. /; Ri x D k jj x i x C j x ij x C j x ji x (16.12) where Z 1 123::: K . /. / . /. / ı n @ . ; / . / j3 ::: jn ; js x D s j x hij3:::j xs Khj x y ^ i y ^ dy dy .n 2/Š n ˙ ÂZ Ã (16.13) h h. /.x/ D dxŒsn2.x; y/ ^ .y/ ^ dx ; ˙

1 k. C 1/ .y x /.y x / k .2 C k/ K .x; y/ l l j j ı s.x; y/; hj D n C lj !n 2.k C 1/ jy xj 2.k C 1/

123:::n ı ::: stands for the generalized Kronecker symbol, denotes the Hodge star hij3 jn X j1 jk j1 jk operator and sk.x; y/ D s.x; y/dx :::dx dx :::dx is the double k-form

j1<:::

j Denoting by .x; y/ D .1j.x; y/;:::;nj.x; y//, j D 1;:::;n, (see (16.3)), and keeping in mind (16.2), we explicitly write the following quantity

Œ j. ; / Li;x x y D Ä 1 2 .1 / . 1/ .x y /.x y / .x y / .x/ C k ı nk C i i j j p p p ij C 2 n !n 2.1 C k/ 2.k C 1/ jx yj jx yj Ä k .2 C k/ .x y / .x/ .x y / .x/ C j j i i i j : (16.16) 2.k C 1/ jx yjn

Hence from (16.16) we get that, if D k=.2 C k/, the kernel of (16.15) has a weak singularity, i.e. Á k=.2Ck/Œ j. ; / O 1nC Li;x x y D jx yj

(otherwise the kernels of (16.15) have a strong singularity). Then the operator Tk=.2Ck/ W ŒLp.˙/n ! ŒLp.˙/n is compact. Thus, we achieve the following claim. Theorem 1 The singular integral operator R can be reduced on the left by Rk=.2Ck/. We deduce that the singular integral system (16.10) admits a solution if and only if Z

i ^ i D 0; i D 1;:::;n; ˙

Œ q .˙/n =. 1/ for every 2 Ln2 (q D p p ) being a solution of the homogeneous adjoint system Z . / . / Œ . ; / 0; ˙; Rj x D i y ^ dy ij x y D a.e. x 2 ˙

i; j D 1;:::;n. We prove (see [CiEtAl11, Th. 5.1]) that R .x/ D 0 if and only if i is a weakly closed .n2/-form. Consequently, the singular integral system R' D df is always solvable. Since the solution of a Dirichlet problem with constant datum on each connected component ˙j of the boundary ˙ can be represented by means of an elastic single- layer potential (see [CiEtAl11, Th. 5.3]), we have our main result (see [CiEtAl11, Th. 5.4]). Theorem 2 If n 3, the Dirichlet problem (16.6) has a unique solution in terms of an elastic single-layer potential (16.4). In particular, its density can be written as ' D '0 C , where '0 satisﬁes the singular integral system (16.10) and is the density of an elastic single-layer potential which is constant on each connected component of ˙. 188 A. Malaspina

Remark on the Two-Dimensional Case. Theorem 2 also holds for n D 2, provided that we give some additional considerations. First, we recall that there are some domains in which not every harmonic function can be represented by means of a harmonic single-layer potential. For instance, on the unit disk we have Z

ln jx yj dsy D 0; jxj <1: jyjD1 Similar domains occur also in elasticity theory, in fact if r D exp.k=.2.k C 2///,on the circle Cr centered at the origin, we achieve (see [CiEtAl11, Lemma 4.3]) Z Z

.x; y/ .1; 0/dsy D .x; y/ .0; 1/dsy D 0; jxj < r: Cr Cr This implies that in the ball of radius r D exp.k=.2.k C 2/// centered at the origin, we cannot represent any smooth solution of the system (16.1) by means of an elastic single-layer potential. If there exists some constant vector which cannot be represented in the simply connected domain ˝ 2 by an elastic single-layer potential, we say that its boundary ˙ is exceptional. In [CiEtAl11, Th. 4.5] we show that also for multiply connected domains we have such particular cases. In particular, we prove that one cannot represent any constant vectors by an elastic single-layer potential and that this happens if and only if the exterior boundary ˙0 D @˝0 is exceptional. Then, when n D 2 and ˙0 is not exceptional, Theorem 2 is also true. Instead, if n D 2 and ˙0 is exceptional, the solution the Dirichlet problem (16.6) can be represented as an elastic single-layer potential plus a vector constant. Remark on the Reduction. The reduction of the singular integral operator R given by (16.11), obtained by means of Rk=.kC2/ (see (16.12) with D k=.k C 2/), is not an equivalent reduction in the usual sense (for this deﬁnition see, e.g., [MiPr86, p. 19]), because N.Rk=.kC2// ¤f0g, N.Rk=.kC2// being the kernel of the operator Rk=.kC2/. However, we still have an equivalence between (16.10) and the Fredholm system Rk=.kC2/R' D Rk=.kC2/df .Infact,asin[Ci92, pp. 253–254], one can prove that N.Rk=.kC2/R/ D N.R/. This implies that if is such that there exists at least a solution of the system R' D , then this system is satisﬁed if and only if Rk=.kC2/R' D Rk=.kC2/ . Since we know that the system R' D df is solvable, we have that R' D df if and only if ' is solution of the Fredholm system Rk=.kC2/R' D Rk=.kC2/df . Lyapunov–Tauber Theorem. In the harmonic potential theory, the Lyapunov– Tauber theorem gives results for the limit on the boundary of the normal derivative for scalar double-layer potential (see, e.g., [Gu67]) and this plays a central role in the study of BVPs with the boundary integral equations methods. Analogous results for a double-layer potential of the classical three dimensional elasticity theory are also known (see, e.g., [KuEtAl79, p. 319]). In [CiEtAl11, Lemma 3.7]) we prove an extension of the Lyapunov-Tauber theorem to n variables for a generalized double- layer potential. 16 An Indirect Boundary Integral Equation Method in Elastostatics 189

Given W ˙ ! n, by the generalized double-layer potential W Œ associated with the kernel L Œ .x; y/ we mean Z Œ . / . . ; //0 . / : W x D Ly x y y d y ˙

Theorem 3 If 2 ŒW1;p.˙/n, then . Œ / . Œ / LC;i W D L;i W . /K . / K . / K . / D k jj d i C j ij d C j ji d ˙ . Œ / . Œ / a.e. on , where LC W and L W denote the internal and the external angular boundary limit (for this deﬁnition see, e.g., [KuEtAl79, p. 293]) of . Œ / K L W , respectively, and ij are given by (16.13). Similar results can be found for thermoelasticity ([KuEtAl79, Na97]), thermo- electromagnetoelasticity ([Mr10]), couple-stress theory ([KuEtAl79, CiEtAl13a]) and for Stokes ﬂow ([CiEtAl13b, Go15]). We note that the proof of Theorem 3 is novel for the use of the theory of differential forms. Traction Problem. Theorem 2 allows to study the eventuality of representing the solution of the traction problem (16.7) in terms of an elastic double-layer potential (16.5) in a multiply connected domain. This is possible if and only if the given forces on every connected components ˙j (j D 0;:::;m) of the boundary ˙ are balanced: Z f ˛ d D 0; j D 0;:::;m: (16.17) ˙j

In fact, if WŒ is an elastic double-layer potential with density 2 ŒW1;p.˙/n, thanks to Theorem 3, the boundary condition LWŒ D f turns into the following system:

R1.d / D f ; where R1 is deﬁned by (16.12) with D 1. In view of Theorem 2,any 2 ŒW1;p.˙/n can be written as Z

Œ .x/ D .x; y/ .y/ dy (16.18) ˙

p n with 2 ŒL .˙/ (keeping in mind that if n D 2 and ˙0 is exceptional, Œ is given by Z 2 Œ .x/ D .x; y/ .y/ dy C c; c 2 /: ˙ 190 A. Malaspina

Since d D R , we infer

R1.d / D R1R. /:

On account of (16.14) with D 1, we have that R1.d / D f is equivalent to the singular integral system 1 .T1/2 f ; 4 C D (16.19) where T1 is the singular integral operator (16.15) with D 1. Therefore, there exists a solution of the traction problem if and only if (16.19) is solvable. Then we outline the next claim (for the details of the proof see [CiEtAl11, Th. 6.2]). Theorem 4 Given f 2 ŒLp.˙/n, the traction problem (16.7) admits a solution (determined up to an additive rigid displacement) in terms of an elastic double- layer potential (16.5)WŒ if and only if f satisfies (16.17). Moreover, its density is written in terms of an elastic single-layer potential Œ given by (16.18), where is a solution of the singular integral system (16.19). Finally, we point out that if f satisfies the only condition (16.8), we need to modify the representation of the solution by adding some extra terms (see [CiEtAl11, Th. 6.4]). Further Applications. Recently, in the same spirit of the above results, we have dealt with other systems of partial differential equations (see contributions for Laplace equation [CiEtAl12], for Stokes system [CiEtAl13b], for Cosserat theory [CiEtAl13a, CiEtAl15a], for viscoelasticity theory [CiEtAl14]), and for second- order divergence form elliptic operators with variable coefficients [CiEtAl15b]).

References

[Ci88] Cialdea, A.: On the oblique derivation problem for the Laplace equation, and related topics. Rend. Accad. Naz. Sci. XL Mem. Mat. 12(1), 181–200 (1988) [Ci92] Cialdea, A: The multiple layer potential for the biharmonic equation in n variables. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 3(4), 241–259 (1992) [CiEtAl11] Cialdea, A., Leonessa, V., Malaspina, A.: Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains. Bound. Value Prob. 2011(53), 25 pp. (2011) [CiEtAl12] Cialdea, A., Leonessa, V., Malaspina, A.: On the Dirichlet and the Neumann problems for Laplace equation in multiply connected domains. Complex Var. Elliptic Equ. 57(10), 1035–1054 (2012) [CiEtAl13b] Cialdea, A., Leonessa, V., Malaspina, A.: On the Dirichlet problem for the Stokes system in multiply connected domains. Abstr. Appl. Anal. 2013, Art. ID 765020, 12 pp. (2013) [CiEtAl13a] Cialdea, A., Dolce, E., Malaspina, A., Nanni, V.: On an integral equation of the ﬁrst kind arising in the theory of Cosserat. Int. J. Math. 24(5), 21 pp. (2013) 16 An Indirect Boundary Integral Equation Method in Elastostatics 191

[CiEtAl14] Cialdea, A., Dolce, E., Leonessa, V., Malaspina, A.: New integral representations in the linear theory of viscoelastic materials with voids. Publ. Math. Inst. Nouvelle série 96(110), 49–65 (2014) [CiEtAl15a] Cialdea, A., Dolce, E., Malaspina, A.: A complement to potential theory in the Cosserat elasticity. Math. Methods Appl. Sci. 38(3), 537–547 (2015) [CiEtAl15b] Cialdea, A., Leonessa, V., Malaspina, A.: The Dirichlet problem for second-order divergence form elliptic operators with variable coefficients: the simple layer potential ansatz. Abstr. Appl. Anal. 2015, Art. ID 276810, 11 pp. (2015) [Fi61] Fichera, G.: Spazi lineari di k-misure e di forme differenziali. In: Proc. Internat. Sumpos. Linear Spaces (Jerusalem 1960), pp. 175–226 (1961) [Fl63] Flanders, H: Differential Forms with Applications to the Physical Science. Academic, New York/San Francisco/London (1963) [Go15] Gonzalez, O.: A theorem on the surface traction field in potential representations of Stokes flow. SIAM J. Appl. Math. 75(4), 1578–1598 (2015) [Gu67] Günter, N.M.: Potential Theory and Its Applications to Basic Problems of Mathemat- ical Physics. Translated from the Russian by John R. Schulenberger. Frederick Ungar Publishing Co., New York (1967) [HsWe08] Hsiao, G.C., Wendland, W.L.: Boundary Intgral Equations. Applied Mathematical Sciences, vol. 164. Springer, Berlin (2008) [Ko05] Kohr, M.: A mixed boundary value problem for the unsteady Stokes system in a bounded domain in n. Eng. Anal. Bound. Elem. 29, 936–943 (2005) [Ko07] Kohr, M.: The Dirichlet problems for the Stokes resolvent equations in bounded and exterior domains in n. Math. Nachr. 280(5–6), 534–559 (2007) [KuEtAl79] Kupradze, V.D., Gegelia, T.G., Baselei˘ svili,˘ M.O., Burculadze, T.V.: Three- Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland Series in Applied Mathematics and Mechanics, vol. 25. North- Holland, Amsterdam (1979) [MaEtAl99] Maremonti, P., Russo, R., Starita, G.: On the Stokes equations: the boundary value problem. In: Advances in Fluid Dynamics, pp. 69–140. Quad. Mat. Aracne, Rome (1999) [MiPr86] Mikhlin, S.G., Prössdorf, S: Singular Integral Operators. Springer, Berlin (1986) [Mr10] Mrevlishvili, M.: Liapunov-Tauber type theorem in thermo-electro-magneto-elasticity theory. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 24, 88–92 (2010) [Na97] Natroshvili, D.: Boundary integral equation method in the steady state oscillation problems for anisotropic bodies. Math. Methods Appl. Sci. 20, 95–119 (1997) [StTa02] Starita, G., Tartaglione, A.: On the traction problem for the Stokes system. Math. Models Methods Appl. Sci. 12, 813–834 (2002) Chapter 17 An Instability Result for Suspension Bridges

C. Marchionna and S. Panizzi

17.1 Introduction

An important issue in the mathematical modeling of suspension bridges is the phenomenon of energy transfer from flexural to torsional modes of vibration along the deck of the bridge. The sudden change from a vertical to a torsional mode of oscillation can be dramatically destructive, see, for example, the collapse of the Tacoma Narrows Bridge. The purpose of our results is to provide a contribution to a recent field of research beginning from [ArEtAl15, BeEtAl15], and summarized in [Ga15], according to which internal nonlinear resonances giving rise to the onset of instability may occur even when the aeroelastic coupling is disregarded. The suspension bridge model (fish-bone model) under consideration has been proposed by K.S. Moore [Mo02], revisited, and somehow simplified in [BeEtAl15]. The dynamics of the midline of the deck, modeled as an Euler-Bernoulli beam of length L and width 2l, is coupled with the elastic response of the suspension cables acting on the side ends of the deck. The cross section of the deck is assumed to be a rigid rod with mass density , length 2l and negligible thickness with respect to l; y.x; t/ is the vertical downward deflection of the midline of the deck with respect to the unloaded state, .x; t/ is the angle of rotation of the deck with respect to the horizontal position.

C. Marchionna () Department of Mathematics, Polytechnic University of Milano, Milano, Italy e-mail: [email protected] Stefano Panizzi Department of Mathematical, Physical and Informatics Sciences, University of Parma, Parma, Italy e-mail: [email protected]

The corresponding PDEs system is 8 < Sytt C EI yxxxx C F .y C l sin /C F .y l sin / D 0 : J tt GJxx C l cos ŒF .y C l sin / F .y l sin / D 0 with hinged boundary conditions:

y.0; t/ D y.L; t/ D yxx.0; t/ D yxx.L; t/ D 0; .0; t/ D .L; t/ D 0:

About the meaning of the constant parameters not yet deﬁned: S is the cross section area, I is the planar second moment of area with respect to the plane y D 0, J is the polar second moment of area with respect to the x-axis, and E and G are, respectively, Young’s modulus and the shear modulus. The restoring force F exerted by the hangers is applied to both extremities of the deck whose displacements from the unloaded state are given by y ˙ l sin .No external forces, except gravity, are taken into account. In the classical slackening regime, the hangers behave as linear springs of elastic constant k >0if stretched and do not exert restoring force if compressed. In this case an analytical expression of F is readily written as follows (see [McEtAl87, Mo02]): C F .r/ D k .r C r0/ r0 ; r0 D Sg=2k; g is the gravity: (17.1)

In this note we consider several generalizations of the slackening regime (assumption (S) below), either for restoring forces having sub-linear growth at inﬁnity or for super-linear growth, such as the forces considered in [McEtAl01]. Since our results aim to describe the onset of torsional instability, we neglect the behavior of the bridge when the torsional angle becomes large, and assume that, at least at the beginning, sin and cos 1. Setting z D l , the PDEs system simpliﬁes to: 8 < Sytt C EI yxxxx C F .y C z/ C F .y z/ D 0; (17.2) : 2 Jztt GJzxx C Sl .F .y C z/ F .y z// D 0

As a first investigation, through a Galerkin projection, we study the interaction between both the first flexural and torsional modes of vibration. Let us assume that the displacements are well approximated by their first mode of vibration, that is:

y.x; t/ ' y1.t/ sin x; z.x; t/ ' z1.t/ sin x; then, dropping the index 1, we reduce the PDEs system to the following ODEs system: 8

EI 4 G 2 Sl2 ˛ D ;ˇD ; D ; (17.4) SL4 L2 J and Z 1 2 f .r/ D F .r sin x/ sin xdx: S 0

It is worth noting that the transform F 7! f falls within the family of Abel transforms, and has a slightly regularizing effect. Moreover f inherits the essential properties of F . The system (17.3) is conservativeR with an energy that, at least for the cases here r considered, is nonnegative ( F.r/ D 0 f .s/ ds):

yP2 zP2 ˛ ˇ E.y; y; z; z/ y2 z2 F.y z/ F.y z/; P P D 2 C 2 C 2 C 2 C C C therefore all its solutions are bounded and globally deﬁned. We are interested in stability/instability properties of pure ﬂexural periodic solutions of (17.3), that is solutions .w.t/; 0/ , where w.t/ solves

wR .t/ C ˛w.t/ C 2f .w.t// D 0:

We always assume that f satisfies the following generalization of the slackening regime. Assumption (S): a) f is an increasing, continuous function such that f .0/ D 0; b) f is piecewise C1: its derivative is continuous with the exception of a finite (eventually empty) set of points r1 < r2 < ::: < rn not including zero in which there exist the finite limits:

lim f 0.r/I ˙ r!ri c) f 0.0/ D m >0; 0 d) f has asymptotically null slope as r !1, i.e. limr!1 f .r/ D 0. In the first part of the paper we present some analytical results for a class of functions f with the same essential properties of the classical slackening regime (17.1), which is a finite limit slope as the displacements at the side edges of the deck y ˙ z get large (see also, for proofs and details, [MaEtAl16]). We set a condition on the parameters (17.4) and the slope M of f at C1 that guarantees instability for the pure flexural solutions, provided the energy is large enough. 196 C. Marchionna and S. Panizzi

Theorem 1 Assume that f satisﬁes

lim f 0.r/ D M >0; (17.5) r!C1 in addition to assumption (S). Let us set s r s ˇ C 2 M ˇ ˇ C 2 M 0 WD ; 1 WD ; q D : ˛ C 2M ˛ ˇ

If ˇ ˇ ˇ 1 ˇ ˇ q C q ˇ >1; D ˇcos 0 cos 1 2 sin 0 sin 1 ˇ then there exists an energy level E0 such that, if E.w.0/; wP .0/; 0; 0/ > E0, the pure ﬂexural periodic solution .w.t/; 0/ is unstable. The last part of this paper shows some numerical experiments in order to compare different models of slackening satisfying the assumptions of Theorem 1 or merely satisfying (S) with a super-linear behavior at C1 , and showing what can happen for “medium level” energy.

17.2 The Isoenergetic Poincaré Map and the Asymptotic Behavior of Its Linearization Lk

In order to prove Theorem 1, we introduce the Poincaré map at a fixed level of energy to reduce by 1 the degrees of freedom, as in [CaEtAl96], and to study the asymptotic behavior of its linearization when the energy tends to infinity. Then the idea, borrowed from [CaEtAl96], is that of studying the linearized system as the energy E !1(see also [GhEtAl01]). The asymptotic system happens to be less regular, but much simpler! Let wk.t/ be the solution of the Duffing equation

wR k.t/ C ˛wk.t/ C 2f .wk.t// D 0; wk.0/ D k >0; wP k.0/ D 0: (17.6)

Under assumption (S), wk.t/ is an even periodic function, with a certain period k and energy Ek. Let us denote by k Df.wk.t/; wP k.t/; 0; 0/ W t 2 Rg; the orbit in the 4-dimensional phase space of .wk;0/. In short, the isoenergetic ﬁrst return Poincaré map Pk around k is deﬁned as follows: Let .y; z/ be a solution with the same energy of .wk;0/, with

y.0/ > 0; yP.0/ D 0; z.0/ D z0; zP.0/ D z1 17 An Instability Result for Suspension Bridges 197 and let T.z0; z1/ be the ﬁrst return time (which exists under some smallness assumptions on .z0; z1/) when the solution .y; z/ crosses the section fPy D 0; y >0g, then

Pk.z0; z1/ D .z.T.z0; z1//; zP.T.z0; z1///:

We linearize the second equation of the ODEs system (17.3) around k and obtain the following Hill equation:: 0 uR.t/ C ˇ C 2 f .wk.t// u.t/ D 0 u.0/ D a; uP.0/ D b (17.7)

The linearized Poincaré map Lk D DPk.0; 0/ is given by Lk.a; b/ D .u.k/; uP.k//, where k is the period of wk.t/. It is the monodromy matrix of the Hill equation, that is Â Ã 0 1 u .k/ u .k/ Lk D 0 1 : (17.8) uP .k/ uP .k/ where u0.t/ and u1.t/ are the two solutions of (17.7) corresponding to the initial conditions .1; 0/ e .0; 1/ respectively. The periodic orbit k is unstable, if the origin is an unstable fixed point of the map Lk. 1 0. / 1. / >1 L Proposition 1 If 2 ju k CPu k j , k has two real eigenvalues 1, 2,such that j1j >1, 2 D 1=1 . Then the periodic orbit k is unstable. If in addition the periodic term of the equation (17.7) is even, the instability test 0 simplifies as ju .k/j >1. In order to study the asymptotic behavior of Lk, we normalize the solution of the Duffing equation (17.6) by setting Wk.t/ D wk.t/=k. The linearized system becomes 8 < f .kWk.t// WR k.t/ C ˛Wk.t/ C 2 D 0; Wk.0/ D 1; WP k.0/ D 0 : k 0 uRk.t/ C .ˇ C 2 f .kWk.t/// uk.t/ D 0 uk.0/ D a; uPk.0/ D b and its limit system, as k !1is 8

0. / v0. / In particular, if uk t and t are the solutions of the Hill equations with data .1; 0/, respectively, of the k-system and the limit system, then

0. / v0. /: uk k ! 1

Now we begin to see where the discriminant in Theorem 1 comes from: ought 0 0 to the simplicity of g.r/, it is possible to compute v .1/ D .Ifjv .1/j >1,the trivial solution of the limit-Hill equation is unstable, and the same happens for the k-Hill equation, for k large enough. Unfortunately, the method does not give any estimate on the size of the critical energy level E0, where this “high energy type” of instability begins to occur.

17.3 Some Related Problems and Questions

We decided to numerically compare a set of 6 functions f in (17.3) modeling the slackening regime, all satisfying the assumption (S). In order to have the same elastic response for small displacement and approximately the same level for the slackening, we chose to ﬁx two parameters

f 0.0/ D m; lim f .r/ Dh; m; h >0: (17.9) r!1

The ﬁrst three functions satisfy the assumption (17.5) with the same slope 0 M D m D f .0/ at C1,aswellasf4 but with a different M D 2m slope at C1. The last two functions do not meet the last condition (17.5) having polynomial and exponential growth at C1, respectively, (an exponential growth was proposed by [McEtAl01]). If we deﬁne the slackening point as r0 D h=m, the chosen model functions are

f1.r/ D mr; r r0; f1.r/ Dh; r Är0; Z =2 4 f2.r/ D f1. r sin x/ sin xdx 0 hr f3.r/ D mr; r 0; f3.r/ D q r <0 2 2 r C r0 q 2 2 f4.r/ D mr C m r C r0 h

n f5;n.r/ D h.1 C r=.nr0// h; r r0; f5;n.r/ Dh; r Är0; .n >1/

r=r0 f6.r/ D h.e 1/: 17 An Instability Result for Suspension Bridges 199

The first function f1 is basically the Moore-McKenna F .r/ in (17.1), and is so simple that allows to explicitly compute, depending on the datum w.0/ D k, the solutions of the linearized problem defined by (17.6) and (17.7), and its instability discriminant. These analytical results have been useful to validate the Matlab numerical results for the whole set of functions. The second function f2 is a rescaling of fQ in [MaEtAl16] (which we refer to for its analytical formula); precisely . / . 4 / f2 r D 4 fQ r , where fQ is the projection on the first eigenmode of (17.1). The 2 function f3 was introduced by [BeEtAl16] and is slightly more regular being C .R/. From a modeling point of view, if we suppose that for larger downward deflection the stiffness has to increase, the last three function may be more suitable, with stronger stiffness behavior from f4 to f6. 0 A first mathematical remark is that the limk!0 u .k/ depends only on the “structural constants” (17.4) and f 0.0/ D m. More precisely, we have p 2 0 2 ˇ C 2 m lim k D p ; lim u .k/ D cos. p /: k!0 ˛ C 2m k!0 ˛ C 2m

So the following proposition holds, due to a continuity argument: p 2 ˇC2 m Proposition 2 If f 0.0/ D m and p … N, then there exists a suitable level of ˛C2m energy E0 such that for lesser energies the linearized Poincaré map is stable. That is, if we ﬁx, for example, ˛, ˇ, and , we have linear stability with the exception of a countable set of values of m. A reasonable conjecture could be that the Poincaré map is stable, or at least linearly stable for every set of parameters, at suitable low energies, for all functions satisfying the (S) assumption. A very interesting result could be to set some upper bound for the energy in order to guarantee at least linear stability.

17.4 Numerical Examples

The following numerical simulations report the behavior of L, half the trace of the monodromy matrix of the Hill equation (17.7), as a function of wk.0/ D k,forthe whole set of functions. In the case when jLj <1the eigenvalues of Lk are unitary conjugate complex numbers, and the origin is an elliptic point of the Poincaré map. In the first example Figure 17.1, the parameters are fixed to ˛ D ˇ D 0:1, D 3, m D h D 1. The asymptotic value of is approximately D0:5988, thus the map is linearly stable for high energies at least for the first four functions. We can note that the first interval of instability starts approximately at the same level k D 1, that is the value of the slackening point r D r0 for all 6 sample functions, while the second instability takes place in a range 5

alpha =0.1 beta = 0.1 gamma = 3 2 2 1 1 L 0 L 0 −1 −1 −2 −2 2 4 6 8 10 12 14 2 4 6 8 10 12 14 k f1 k f2 2 2 1 1

L 0 L 0 −1 −1 −2 −2 2 4 6 8 10 12 14 2 4 6 8 10 12 14 k f3 k f4 2 2 1 1

L 0 L 0 −1 −1 −2 −2 2 4 6 8 10 12 14 2 4 6 8 10 12 14 k f5 k f6

Fig. 17.1 Half the trace of the monodromy vs k. ˛ D ˇ D 0:1, D 3, m D h D 1, n D 2 in f5;n fourth function which has slope 2m at infinity, and no other instability arises for the last two. We can note also that the behavior of the discriminant L is smoothed out as the functions fi gain regularity. In our second example Figure 17.2, the parameters are fixed to ˛ D 0:1, ˇ D 4, D 3, m D 1, h D 2. Again Theorem 1 tells us that the map is linearly stable for high energies at least for the first four functions. The effect of slackening is clear in every case at k D 2 D r0, even though it is a little anticipated for f4 and f6. We can see that it does not necessarily brings instability, and that for larger k the behavior of L might be quite different, in particular for the last two functions. In conclusion in all the numerical simulations we performed, at least for r0 not too large, the evolution of L (linear stability) was very similar for values of k smaller than the slackening point r0. Instabilities may occur for greater values of k but, of course, the last two models may behave quite differently. Now we comment more accurately Figure 17.3, showing for the function f2 the behavior of the nonlinear isoenergetic Poincaré map defined in Section 17.2.The following simulations are calculated using the Matlab solver ode23t, keeping track of the decay of energy, which is about of 0.001% in sixty interactions. The study of the linearized map shows us that there are two intervals of instability, corresponding to approximately 11and with doubled period when L < 1. In the case L < 1, 17 An Instability Result for Suspension Bridges 201

alpha =0.1 beta = 4 gamma = 3 2 2 1 1 0 0 L L −1 −1 −2 −2 2 4 6 8 10 12 14 2 4 6 8 10 12 14 k f k f 1 2 2 2 1 1 L L 0 0 −1 −1 −2 −2 2 4 6 8 10 12 14 2 4 6 8 10 12 14 k f 3 k f 2 2 4 1 1 L 0 L 0 −1 −1 −2 −2 2 4 6 8 10 12 14 2 4 6 8 10 12 14 k f k f 5 6

Fig. 17.2 Half the trace of the monodromy vs k. ˛ D 0:1, ˇ D 2, D 3, m D 1, h D 2, n D 2 in f5;n towards the end of the first interval of instability, the stable periodic points stay approximately in the same place, while another couple of unstable periodic points with doubled period appear near the origin and collapse in the origin at the end of the interval of instability. In the case L >1it seems that there is only the couple of stable periodic points, that in the end collapses in the origin. In Figure 17.3 each blue small circle represents an interaction of the Poincaré map Pk. The red stars correspond to stable periodic points, the yellow ones to unstable periodic points. The first three subfigures are related to the first interval of instability, the last one to the second. The evolution of the plot of the map in the second interval is indeed much simpler: the same pattern, from smaller to big, and again to small. As a note, we choose to represent the Poincaré map Pk, because of the nice “symmetry” with respect to the vertical axis. The placing of the stable periodic points at the same level of energy has been computed using the twin Poincaré map PE where the manifold E.y; yP; z; zP/ D E intersects the hyperplane y D 0, under the condition yP >0: the initial value problem for the Duffing equation (17.6) becomes p wR E.t/ C ˛wE.t/ C 2f .wE.t// D 0; wE.0/ D 0; wP E.0/ D 2E >0 and Proposition 1 has to be used in its more general form. 202 C. Marchionna and S. Panizzi

Energy=1,27 Energy=2.36 1 y(0) =1.1 y(0) =1.5 Dy(0) =0 2 Dy(0) =0 z(0) =0 z(0) =0 0.5 Dz(0) =0.01 Dz(0) =0.01 1

0 0 Dz Dz

-1 -0.5 yP(0) =1.0708 yP(0) =1.3543 DyP(0) =0 DyP(0) =0 zP(0) =0 -2 zP(0) =0 DzP(0) =0.633 DzP(0) =1.6166 -1 -2 -1 0 1 2 -2 -1 0 1 2 z z

Energy=5.91 Energy=105 1 4 y(0) =2.3752 y(0) =10 Dy(0) =0 Dy(0) =0 z(0) =0 z(0) =0 Dz(0) =0.01 Dz(0) =0.01 2 0.5

0 0 Dz Dz

-2 yP(0) =2.0923 yPI(0) =2.37 -0.5 yP(0) =9.85 DyP(0) =0 DyPI(0) =0 DyP(0) =0 zP(0) =0 zPI(0) =0.11 zP(0) =1.7527 -4 DzP(0) =2.8059 DzPI(0) =0 DzP(0) =0 -1 -2 -1 0 1 2 -2 -1 0 1 2 z z

Fig. 17.3 Evolution of the nonlinear Poincaré map for function f2, ˛ D ˇ D 0:1, D 3, m D h D 1. In any subﬁgure (different levels of energy) the initial data for the interaction represented by blue circles can be found in the top left, the initial data for a periodic stable solution in the bottom left and the initial data for a periodic unstable solution in the bottom right

References

[ArEtAl15] Arioli, G., Gazzola, F.: A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge. Appl. Math. Model. 39, 901–912 (2015) [BeEtAl16] Benci, V., Fortunato, D., Gazzola, F.: Existence of torsional solitons in a beam model of suspension bridge. Arch Rational Mech Anal 1–27 (2017). doi:10.1007/s00205- 017-1138-8 [BeEtAl15] Berchio, E., Gazzola, F.: A qualitative explanation of the origin of torsional instability in suspension bridges. Nonlinear Anal. 121, 54–72 (2015) [CaEtAl96] Cazenave, T., Weissler, F.B.: Unstable simple modes of the nonlinear string. Q. Appl. Math. 54, 287–305 (1996) [Ga15] Gazzola, F. : Mathematical Models for Suspension Bridges: Nonlinear Structural Instability. MS&A. Modeling, Simulation and Applications, vol. 15. Springer, Cham (2015) [GhEtAl01] Ghisi, T., Gobbino, M.: Unstable simple modes for a class of Kirchhoff equations. Ann. Fac. Sci. Toulouse Math. (6), 10, 639–658 (2001) 17 An Instability Result for Suspension Bridges 203

[MaEtAl16] Marchionna, C., Panizzi, S.: An instability result in the theory of suspension bridges. Nonlinear Anal. 140, 12–28 (2016) [McEtAl01] McKenna, P.J., Tuama, C.Ó.: Large torsional oscillations in suspension bridges visited again: vertical forcing creates torsional response. Am. Math. Mon. 108, 738–745 (2001) [McEtAl87] McKenna, P.J., Walter, W.: Nonlinear oscillations in a suspension bridge. Arch. Ration. Mech. Anal. 98, 167–177 (1987) [Mo02] Moore, K.S.: Large torsional oscillations in a suspension bridge: multiple periodic solutions to a nonlinear wave equation. SIAM J. Math. Anal. 33, 1411–1429 (2002) Chapter 18 A New Diffeomorph Conformal Methodology to Solve Flow Problems with Complex Boundaries by an Equivalent Plane Parallel Problem

A. Meneghetti, B.E.J. Bodmann, and M.T. Vilhena

18.1 Introduction

The motivation for the present contribution are flow problems that may have boundaries with complex geometry. Conventional approaches use typically step wise approximations that allow to cast the problem in schemes for established numerical solvers. The present approach is for non-plane but continuous boundaries with some restrictions to be indicated below. More specifically, the problem with curvilinear boundaries is transformed into a problem with plane parallel boundaries by a diffeomorph conformal coordinate transformation. Consequently the operators of the dynamical equations change according to the additional terms from the affine connection. A case study that sketches the novel methodology together with a numerical simulation is presented.

18.2 Coordinate Transformations

In order that a mathematical description of a phenomenon be universal, the dynamical equations shall be independent of the speciﬁc choice of any coordinate system. Thus, in general one can assign different but equivalent coordinate systems to an n-dimensional space. Given two coordinates systems for the same space, a coordinate transformation is a rule that relates the two systems. For example, if X and Y are two coordinates systems of the vector space n and a point P 2 ½ Â n having the representation P.x1;:::;xn/ in the X coordinate system,

A. Meneghetti () • B.E.J. Bodmann • M.T. Vilhena Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil e-mail: [email protected]; [email protected]; [email protected]

© Springer International Publishing AG 2017 205 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_18 206 A. Meneghetti et al. then the coordinate transformation deﬁned by T,

T W yi D yi.x1;:::;xn/ i D 1;2;:::;n; (18.1) is a function T W n ! n that determines the representation of P in the Y coordinate system from the point P representation of the coordinate system X.Of special interest are invertible transformations, so that we can ﬁnd the representation of P in the Y coordinate system from the X coordinate system and vice versa. To ensure that the coordinate transformation T has an inversion in a region ½ it is necessary to require that yi,in(18.1), are continuous, have continuous derivatives and for any point P 2 ½ the determinant of the Jacobian matrix T in ½ is non-zero. ˇ ˇ ˇ@ i ˇ ˇ y ˇ 1 2 n jJT.x/j D ˇ .x/ˇ ¤ 0; 8x D .x ; x ;:::;x / 2 ½ (18.2) @xj

As elaborated in detail in reference [So64], these conditions are sufﬁcient to ensure there is an inverse transformation, (18.3), and that the functions xi are of class C1.D/.

T1 W xi D xi.y1; y2;:::;yn/; for i D 1;2;:::;n (18.3)

By hypothesis, the function yi.x/ in (18.1) and its derivatives are continuous in ½. 1 n If x0 D .x0;:::;x0/ 2 ½, then by Taylor’s formula, there is a local linear approach that can be expressed as

Xn @ i i i y j j y .x/ D y .x0/ C .x x /: @xj 0 jD1

Then the transformation (18.1) is locally linear. As jJT.x0/j¤0 then the system of linear equations has a unique solution. The theorem of inverse functions (2) ensures that, under these conditions, we have a local diffeomorphism. The proof of this theorems (1) and (2) can be found at [Li00, Li11]. Theorem 1 Let ½ Â n be an open and T W ½ ! n defined in (18.1). Suppose @yi ½ the partial derivatives @xi are continuous in , then T is continuously differentiable. Theorem 2 (The Application Inverse Theorem) Let T W n ! n be continuously differentiable on some open set containing x0, and suppose jJT.x0/j¤0. Then there is some open set containing x0 and an open containing T.x0/ such that T W ! is a diffeomorphism. Definition 1 The transformations of the form (18.1) in which yi are continuous together with their first partial derivatives and satisfying (18.2) are called admissible transformations. 18 Diffeomorph Conformal Methodology 207

18.3 Transformation by Invariance

Let C W ½ n ! be a real function and a point P 2 ½, with ½ an open domain. Suppose C is continuous at ½ and it is covered by a coordinate system X. The scalar value C.P/ depends on the point P, but it does not depend on the coordinate system in which P is represented. In the coordinate system X, C.P/ should take the form f .x/, where x D .x1; x2;:::;xn/. If we introduce another coordinate system Y with coordinates y D .y1; y2;:::;yn/, using an admissible transformation T,

T W n ! n ; T W xi D xi.y1; y2;:::;yn/; i D 1;2;:::;n ; the scalar function C.P/ can be obtained in the coordinate system Y,by(18.4). f x1.y1;:::;yn/; x2.y1;:::;yn/;:::;xn.y1;:::;yn/ D g.y1;:::;yn/ (18.4)

Writing x D .x1; x2;:::;xn/ and y D .y1; y2;:::;yn/ the relation (18.4) can assume the compact form

f .x.y// D g.y/:

The point P in a coordinate system X is represented by P.x/, in coordinate system Y the same point P is represented by P.y/. In analogy, for the function C applied to point P, that is C.P/,thevalueC.P/ in coordinate system X is determined by f .x/ and in coordinate system Y is determined by g.y/. The transformations change the coordinate system but do not change the result of the scalar function, in other words leave the physics of the phenomenon unchanged. Hence transformations of type (18.4) are called transformations by invariance. Let T1, T2 and T3 be admissible transformations, where

1 T1 W y D y.x/ ! T1 W x D x.y/; 1 T2 W z D z.y/ ! T2 W y D y.z/; and T3 D T2T1,

1 T3 W z D z.y.x// ! T3 W x D x.y.z//:

Note that a composition of admissible transformations results in another admissible transformation. Consider C.p/ a scalar, which in the coordinate system X is represented by f .x/. In the coordinate system Y the scalar C.p/ is represented by g.y/ and g.y/ is determined by

G1 W g.y/ D f .x.y//: 208 A. Meneghetti et al.

Another possible representation of the scalar C.p/ can be given in coordinate system Z, determined by the function h.z/. The function h.z/ is obtained by

G2 W h.z/ D g.y.z//:

However, using the product transformation T3 D T2T1 we get

G3 W h.z/ D f .x.y.z///:

Note that G3 D G2G1. The relationship between the transformations T and G is such that the product of two transformations T2T1 correspond to the product of the transformations G2G1. When this relationship is obtained between two groups T and G, the groups are called isomorphic. The isomorphism between T and G is an important characteristic of a class of invariants called tensors. In fact, physically relevant quantities have generally tensor representation. While tensor theory involves geometric entities of general and abstract character in the following we make use of a particular application of tensor theory.

18.4 Application to a Navier-Stokes Problem

The Navier-Stokes equation can be simpliﬁed to the case of incompressible ﬂow (density is constant) according to equation (18.5) in tensor representation [Sc79], where we tacitly used the Einstein sum convention. Á Á 2 @uk i @uk k @P @ uk C u D X C 2 @t @xi @xk @xi (18.5) @uk 0 @xk D

Here ui are the three orthogonal velocity components (k; i 2f1; 2; 3g), Xk is an external source term, P is the local pressure, the density and the dynamic viscosity. Now, consider an admissible transformation T, that establishes a relation between the Cartesian coordinate system x1 x2 x3 and the generalized coordinate system 1 2 3.

T W xi D xi.1;2;3/; for i D 1; 2; 3: (18.6)

Upon applying the transformation, equation (18.5) becomes (18.7). 18 Diffeomorph Conformal Methodology 209 Á k k @j @u C ui @u D @t @j @xi Á 2 2 k @P @j @ l @uk j @j @l @ uk X C 2 ı C (18.7) @j @xk @xi @l i @xi @xi @j@l @uk @j 0 @j @xk D

; ; ; 1; 2; 3 ıj 1 where k i j l 2f g and the Kronecker symbol i D for i D j and zero otherwise.

18.5 Application to an Advection-Diffusion Problem

Consider the advection-diffusion equation deﬁned in (18.8).

2 @C i @C @K @C @ C C u D C K 2 (18.8) @t @xi @xi @xi @xi

In (18.8), C D C.fxig/ is the pollutant concentration, K D K.fxig/ is the eddy diffusion coefﬁcient and ui D ui.fxig/ are the three orthogonal velocity components. Upon applying an admissible transformation (18.6), equation (18.8) becomes (18.9).

@j @C ui @C @t C @j @xi D Á 2 @K @l @C @j @ j @C j @j @l @2C (18.9) C K 2 ı C @l @xi @j @xi @xi @j i @xi @xi @j@l

18.6 A Numerical Example

Any specific application with its in general complex curved boundaries implies in a specific transformation, that transforms a domain bounded by a combination of curved surfaces into an equivalent plane parallel problem. To this end the equations (18.7) and (18.9) are admissibly transformed equations with representation in a generalized coordinate system. Since the present discussion is to show by a simple example how the proposed solution method works, we consider a flow problem in a two-dimensional geometry, where the lower boundary is shaped by the presence of a hill (see Figure 18.1). For this problem we determine the solution of the transformed as well as the original problem. The physically relevant domain lies in a rectangular area with x 2 Œ0; 6000 m and z 2 Œ0; 2090 m, where the latter is a typical value for the planetary boundary layer height. The relief of the hill obeys the equation

2 z D A exp..x x0/ =B/; 210 A. Meneghetti et al.

z 2000

1500

1000

500

x 1000 2000 3000 4000 5000 6000

Fig. 18.1 Two-dimensional domain with curved lower boundary

2 where we adopt A D 200 m, B D 50000 m and x0 D 1300 m. An effluent is released from a high source at H D 115 m without any thrust and by the presence of a wind blowing in the horizontal dimension. The pollutant is emitted with constant intensity, where the horizontal velocity u has little variation and thus will be approximated constant, while the vertical velocity w is assumed zero. The horizontal and vertical turbulent diffusivity vary with height above ground as well as the horizontal distance the pollutant is transported. We want to determine the time dependent concentration of pollutant C D C.x; z; t/ and its dispersion in two- dimensional space. Recalling, the two-dimensional advection-diffusion equation for atmospheric flow problems satisfies equation (18.10). Â Ã Â Ã @C @C @C @ @C @ @C C u C w D K C K (18.10) @t @x @z @x @x @z @z

Here C D C.x; z; t/ is the pollutant concentration in units of g=m3, K D K.x; z/ is the turbulent diffusivity in units of m2=s and u D u.x; z/ and w D w.x; z/ are the horizontal and vertical velocity component in units of m=s. In this problem we assume the initial conditions and boundaries deﬁned in equations (18.11) and (18.12), respectively.

C.x; z;0/D 0 8x 2 Œ0; 6000 m and 8z 2 Œ0; 2090 m (18.11)

.0; ; / 40 g ı. 115 / Œ0; 2090 Œ0; / C z t Dˇ 3 z m 8z 2 m and 8t 2 1 ˇ m @C .x; z; t/ˇ D 0 8x 2 Œ0; 6000 m and 8t 2 Œ0; 1/ @z ˇ 2090 ˇzD m (18.12) @C .x; z; t/ˇ D 0 8z 2 Œ0; 2090 m and 8t 2 Œ0; 1/ @x ˇ 6000 ˇxD m @C . ; ; / 0 Œ0; 6000 Œ0; / @ x z t ˇ D 8x 2 m and 8t 2 1 z zD0

The ﬁrst step of the proposed solution method starts with the coordinate transformation such that the new domain is plane parallel. Let the transformation be 18 Diffeomorph Conformal Methodology 211 ( D .x; z/; T W (18.13) D .x; z/:

The conversion shall be an admissible transformation such as (18.2), therefore (18.13) must satisfy (18.14). ˇ ˇ ˇ ˇ ˇ @ @ ˇ . ; / ˇ @x @z ˇ 0; . ; / Œ0; 6000 Œ0; 2090 / jJT x z jDˇ @ @ ˇ ¤ 8 x z 2 m m (18.14) @x @z

Next, using tensor notation the advection-diffusion equation (18.10) can be represented in the generalized coordinate system (18.15), where all variables were explained in (18.17).

@j @C ui @C @t C @j @xi D Á 2 @K @l @C @j @ l @C j @j @l @2C (18.15) C K 2 ı C @l @xi @j @xi @xi @l i @xi @xi @j@l

The operational form for computing is given in (18.16). Á Á @ @ @C C u @C C @ @C C w @C C @ @C D @t @x @ @x @Á @z @ Á @z @ @ @ @K C @K @ @C C @C @ C @ @x @ @x Á@ @x @ @x Á @ @ @K C @K @ @C C @C @ C @ @z @ @z @ @z @ @z 2 @ @C @ @ @2C @2 @C @ @ @2C (18.16) K @x2 @ C @x @x @2 C @x2 @ C @x @x @@ C 2 @ @C @ @ @2C @2 @C @ @ @2C @x2 @ C @x @x @@ C @x2 @ C @x @x @2 2 @ @C @ @ @2C @2 @C @ @ @2C @z2 @ C @z @z @2 C @z2 @ C @z @z @@ CÁ 2 @ @C @ @ @2C @2 @C @ @ @2C @z2 @ C @z @z @@ C @z2 @ C @z @z @2

For this example we use the diffusion coefﬁcients K found by the authors of references [PuEtAl13, NuDe13].

2 Â 1 2 Ã 3 3 3 0:38K1 K2 1C0:75K1 K3 K2 .0:31/.2090/ .; / K D Â 1 1 Ã2 3 3 1 0:82K3 C1:24K1 K3 K2 Á 2 2 2 3 .; / 1 3 0:75 K1 D 2090Á 116 C 0:31 K2.; / D .3:6/.2090/ .; / 1 4 0:0003 8 K3 D exp 2090 exp 2090 212 A. Meneghetti et al.

z 2000

1500

1000

500

x 1000 2000 3000 4000 5000 6000

Fig. 18.2 Curvilinear meshes in the coordinate system x z

In order to show some properties of the transformation, Figure 18.2 represents a curvilinear grid that is stretched locally in the vertical direction such that the new physical domain is plane parallel. The curvilinear grid proﬁle to be deformed is given by

. /2 x x0 zb D .1 b/Ae B C 2090b ; b 2 Œ0; 1 :

A change in the coordinates may now be established according to the grid-point scheme shown in Figure 18.2. In the present example, far from the elevation the original coordinates and the transformed ones are almost identical, whereas around 2 x0 with z A.1 .x x0/ =B/ the coordinates differ. 8 < D x Â Ã 2 T W .xx0/ (18.17) : 1 z B D z 2090 m Ae

We observe that equation (18.17) satisﬁes the condition (18.14). ˇ ˇ ˇ ˇ ˇ 10ˇ 2 2 jJT.x; z/jDˇ .xx / .xx / ˇ ¤ 0; 2.xx0/ 0 1 0 ˇ 1 z B 1 B ˇ 2090 m B Ae C 2090 m Ae

8.x; z/ 2 Œ0; 6000 m Œ0; 2090 m. Now, following the prescription in reference [BoEtAl15], we solve the equation (18.16) subject to the initial (18.11) and boundary conditions (18.12). The obtained solution in the generalized coordinate system is shown in Figure 18.3. The inverse transformation generates the solution in the original coordinate system x z shown in Figures 18.4 and 18.5, where the latter is a zoom in the domain region x 2 Œ0; 3000 m and z 2 Œ0; 500 m with the most signiﬁcant changes in the dispersion process. 18 Diffeomorph Conformal Methodology 213

Fig. 18.3 Pollutant concentration in the coordinate system

Fig. 18.4 Pollutant concentration in the coordinate system x z

Fig. 18.5 Zoom of the pollutant concentration in the coordinate system x z into the region with signiﬁcant changes

18.7 Conclusion

The resolution methodology presented suggests that in order to resort to analytical or semi-analytical approaches a class of problems can be studied using a generalized coordinate system, that allows to transform a dynamical problem with complex geometries as boundaries into a problem with plain parallel boundaries but at the cost of adding terms to the differential operators of the equation, due to the affine connection. However, the solution of the original operator could still be used in a recursive system that makes use of the decomposition method where the correcting terms due to the affine connection enter only with homogeneous boundary conditions which we address in a follow-up work. The use of an 214 A. Meneghetti et al. appropriate coordinate system is not new, in standard problems in physics and engineering frequently polar, cylindrical and spherical coordinates are used to better adapt the geometry to the problem in consideration. As a generalization one could think of the boundary geometry of a given problem as the indicator as to what is the most adequate coordinate system in order to solve the problem at hand. The authors of this work are completely aware of the limitations of this methodology due to the restrictions imposed by the transformation properties. However, numerical approaches for the type of problems considered in the present discussion in general are easier to implement computationally if one has the computer power at hand. Even then severe problems arise, especially when irregular meshes are used, where it is not trivial at all to correct for violation of fundamental conservation laws. It is noteworthy that the way we introduced the coordinate transformations in the present work, these transformations are manifest covariant and thus preserve symmetries, i.e. conservation laws. Moreover, once an analytical representation of the solution is found, computational issues for simulations should also be faster; what processing time is concerned and moreover less hardware power should be of need. Future studies will shed light on these potential advantages. A further study will also focus on the question of defining formally the class of problems that can be successfully studied using the proposed methodology.

References

[BoEtAl15] De Bortoli, A.L., Andreis, G.S.L., Pereira, F.N.: Modeling and Simulation of Reactive Flows. Elsevier, Burlington (2015) [Li00] Lima, E.L.: Curso de Análise Volume 2. IMPA, Rio de Janeiro (2000). [Li11] Lima, E.L.: Variedades Diferenciáveis. IMPA, Rio de Janeiro (2011) [NuDe13] Nunes, A.B., Degrazia, G.A.: Turbulent eddy diffusivities for a planetary boundary layer generated by thermal and mechanical effects. Rev. Bras. Geofísica 31, 609–618 (2013) [PuEtAl13] Puhales, F.S., Rizza, U., Degrazia, G.A., Acevedo, O.C.: A simple parameterization for the turbulent kinetic energy transport terms in the convective boundary layer derived from large eddy simulation. Physica A 392, 583–595 (2013) [Sc79] Schlichting, H.: Boundary-Layer Theory. McGraw-Hill, New York (1979) [So64] Sokolnikoff, I.S.: Tensor Analysis. Wiley, London/New York (1964) Chapter 19 A New Family of Boundary-Domain Integral Equations for the Mixed Exterior Stationary Heat Transfer Problem with Variable Coefﬁcient

S.E. Mikhailov and C.F. Portillo

19.1 Introduction

In this paper, a mixed boundary value problem for the stationary heat transfer partial differential equation with variable coefficient in an exterior domain is reduced to a system of direct segregated parametrix-based boundary-domain integral equations. We use the parametrix different from the one employed in [Mi02, CMN09, CMN13] but coinciding with the one implemented in [MiPo15a] for interior domains. It leads to a parametrix-based integral potentials involving the variable PDE coefficient that depends on the variable of integration. This makes the relations between the parametrix-based potentials and their counterparts for constant coefficients more complicated than in [Mi02, CMN09, CMN13]. Notwithstanding, the mapping properties in weighted Sobolev spaces similar to the ones in [CMN13] still hold. Therefore, it is possible to prove equivalence and invertibility for a Boundary-Domain Integral Equation system derived from the original boundary value problem. Unlike for the case of bounded domains, the Rellich compactness embedding theorem is not available for Sobolev spaces defined over unbounded domains. Nevertheless we were able to prove that the remainder operator is compact. Therefore, we can still benefit from the Fredholm alternative theory to prove the BDIE system operator invertibility.

S.E. Mikhailov () Brunel University West London, Uxbridge, UK e-mail: [email protected] C.F. Portillo Oxford Brookes University, Wheatley, UK e-mail: [email protected]

19.2 Preliminaries

C Let ˝ D ˝C be a unbounded exterior connected domain, ˝ WD 3 X ˝ the complementary (bounded) subset of ˝. The boundary S WD @˝ is simply 1 connected, closed and inﬁnitely differentiable, S 2 C . Furthermore, S WD SN [ SD where both SN and SD are non-empty, connected disjoint subsets of S. The border of 1 these two submanifolds is also inﬁnitely differentiable: @SN D @SD 2 C . We consider the following partial differential equation:

3 Â Ã X @ @u.x/ Au WD A.x/Œu.x/ WD a.x/ D f .x/; x 2 ˝; (19.1) @x @x iD1 i i where u.x/ is the unknown function, a.x/ 2 C1, a.x/>0, is a variable coefﬁcient, and f is a given function on ˝. It is easy to see that if a Á 1, then the operator A becomes the Laplace operator . s s s s In what follows, H .˝/ D H2.˝/, H .S/ D H2.S/ denote the Bessel potential spaces (coinciding with the Sobolev–Slobodetski spaces if s 0). For an open set ˝ D.˝/ 1 .˝/ D.˝/ , we denote D Ccomp , is the Schwartz space of sequentially continuous functionals on D.˝/, while D.˝/ is the set of restrictions on ˝ of 3 es s functions from D. /. We also denote H .S1/ Dfg W g 2 H .S/; supp g S1g, s. / s. / H S1 DfrS1 g W g 2 H S g, where S1 is a proper submanifold of a closed surface

S and rS1 is the restriction operator on S1. 2 1 Let !.x/ D .1 Cjxj / 2 be the weight function and let

1 1 L2.!I ˝/ WD fg W !g 2 L2.˝/g; L2.! I ˝/ WD fg W ! g 2 L2.˝/g be the weighted Lebesgue spaces and H 1.˝/ be the weighted Sobolev (Beppo- Levi) space,

1 1 H .˝/ WD fg 2 L2.! I ˝/ Wrg 2 L2.˝/g; 2 !1 2 2 ; kgkH1.˝/ WD k gkL2.˝/ CkrgkL2.˝/ cf. [Gr78, Ha71, CMN13] and references therein. Let us also define as He 1.˝/ a completion of D.˝/ in H 1.3/, while He 1.˝/ WD ŒH 1.˝/, H 1.˝/ WD ŒHe 1.˝/ are the corresponding dual spaces. The operator A acting on u 2 H 1.˝/ is well defined in the distribution sense as long as the variable coefficient a.x/ is bounded, i.e. a 2 L1.˝/,as

hAu;viDharu; rviDE.u;v/ 8v 2 D.˝/: Z E.u;v/WD E.u;v/.x/dxI E.u;v/.x/ WD a.x/ru.x/rv.x/: (19.2) ˝ 19 A New Family of BDIEs 217

Note that the functional E.u;v/W H 1.˝/He 1.˝/ ! is continuous, thus by the density of D.˝/ in He 1.˝/, also is the operator A W H 1.˝/ ! H 1.˝/ (19.2) which gives the weak form of the operator A in (19.1). From now on, we will assume a.x/ 2 L1.˝/ and that there exist two positive constants, C1 and C2, such that:

The trace operators on S from ˝˙ are denoted by ˙ and the operators ˙ W 1 1 1 1 H .˝˙ ! H 2 .S/ and ˙ W H .˝˙ ! H 2 .S/ are continuous (see, for example, [McL00, Mi11, CMN13]). For u 2 Hs.˝/; s >3=2, we can deﬁne by T˙ the conormal derivative operator acting on S understood in the classical sense:

3 Â Ã Â Ã X @u @u.x/ T˙Œu.x/ WD a.x/n .x/ ˙ D a.x/ ˙ ; (19.4) i @x @n.x/ iD1 i where n.x/ is the exterior unit normal vector to the domain ˝ at a point x 2 S. However, for u 2 H 1.˝/ (aswellasforu 2 H1.˝/), the classical conormal derivative operator may not exist in the trace sense. We can overcome this difﬁculty by introducing the following function space for the operator A,(cf.[CMN13, Gr78, Ha71])

H 1;0.˝I A/ WD fg 2 H 1.˝/ W Ag 2 L2.!I ˝/g endowed with the norm

2 2 !A 2 : k g kH1;0.˝IA/WDk g kH1.˝/ Ck g kL2.˝/

1;0 1 Now, if u 2 H .˝; A/ we can deﬁne the conormal derivative TCu 2 H 2 .S/ as hinted by the Green’s formula, cf. [McL00, CMN13], Z 1 C ; Œ. C !/A . ; C / 2 . /; hT u wiS WD ˙ 1 u C E u 1w dxI for all w 2 H S ˝˙

1 1 C 2 . / H .˝/ where 1 W H S ! is a continuous right inverse to the trace operator C 1 1 W H .˝/ ! H 2 .S/, whereas the brackets hu;viS represent the duality brackets 1 1 2 of the spaces H 2 .S/ and H 2 .S/ which coincide with the scalar product in L .S/ when u;v 2 L2.S/. 1;0 1 The operator TC W H .˝I A/ ! H 2 .S/ is bounded and gives a continuous extension on H 1;0.˝I A/ of the classical conormal derivative operator (19.4). We C C remark that when a Á 1, the operator T becomes T , which is the continuous 1;0 extension on H .˝I / of the classical normal derivative operator @n WD n r. 218 S.E. Mikhailov and C.F. Portillo

In a similar manner as in the proof of Lemma 4.3 in [McL00] or Lemma 3.2 in [Co88], the ﬁrst Green identity holds for u 2 H 1;0.˝I A/, cf. Eq. (2.8) in [CMN13], Z C C 1 hT u; viS D ŒvAu C E.u;v/dx 8v 2 H .˝/: (19.5) ˝

Applying identity (19.5)tou;v 2 H 1;0.˝I A/ and then exchanging roles and subtracting the one from the other, we arrive to the following second Green identity, (see, for example, Eq. (2.9) in [CMN13]), Z Z ŒvAu uAv dx D Cv TCu CuTCv dS.x/: (19.6) ˝ S

19.3 Boundary Value Problem

We aim to derive a system of Boundary-Domain Integral Equations (BDIEs) equivalent to the following mixed boundary value problem deﬁned in an exterior domain ˝. Find v 2 H 1;0.˝I A/ such that:

Au D f ; in ˝I (19.7)

C ; rSD u D 0 on SDI (19.8)

C ; rSN T u D 0 on SN I (19.9)

2 1 1 where f 2 L .!; ˝/, 0 2 H 2 .SD/, and 0 2 H 2 .SN /. Let us denote the left-hand side operator of the mixed problem as

1 1 1;0 2 2 2 AM W H .˝I A/ ! L .!; ˝/ H .SD/ H .SN /: (19.10)

By using variational settings and the Lax Milgram lemma, it is proved (see Theorem A.6 in [CMN13]) that the operator (19.10) is continuously invertible and thus the unique solvability of the BVP (19.7)–(19.9) follows.

19.4 Parametrices and Remainders

Boundary Integral Equations (BIEs) are derived from BVPs with constant coef- ﬁcients using an explicit fundamental solution. Although a fundamental solution may exist for the variable coefﬁcient case, it is not always available explicitly. 19 A New Family of BDIEs 219

Therefore, we introduce a parametrix or Levi function (see [CMN09, CMN13, MiPo15a, MiPo15b] for more details). In this chapter, we will use the same parametrix as in [MiPo15b],

1 1 3 P.x; y/ D P.x y/ D ; x; y 2 ; a.x/ 4 a.x/jx yj whose corresponding remainder is

X3 Â Ã @ 1 @a.x/ 3 R.x; y/ D P.x; y/ ; x; y 2 : @x a.x/ @x iD1 i i

Condition 1 To obtain boundary-domain integral equations, we will assume the following condition further on, unless stated otherwise:

a 2 C1.3/ and !ra 2 L1.3/: (19.11)

19.5 Surface and Volume Potentials

Here, we present the surface and volume potential type operators which will be involved in the BDIEs derived later on. We provide below the key mapping properties needed to prove the equivalence and invertibility theorems at the end of the chapter. The mapping properties are presented in weighted Sobolev spaces. The analogous properties for the bounded domain case in standard Sobolev spaces can be consulted in [MiPo15b]. The parametrix-based single layer and double layer surface potentials are deﬁned for y 2 3 W y … S,as Z V.y/ WD P.x; y/.x/ dS.x/ y … S; ZS . / C . ; /. / . / : W y WD Tx P x y x dS x y … S S We also deﬁne the following pseudo-differential operators associated with direct values of the single and double layer potentials and with their conormal derivatives, for y 2 S, Z V.y/ WD P.x; y/.x/ dS.x/; ZS W. / C . ; /. / . /; y WD Tx P x y x dS x S 220 S.E. Mikhailov and C.F. Portillo

W0. / ˙ŒV. /; y WD Ty y L˙. / ˙ŒW. /: y WD Ty y Let us introduce now the parametrix-based Newtonian and remainder volume potentials which are deﬁned, for y 2 3,as Z P.y/ WD P.x; y/.x/ dx; ˝ Z R.y/ WD R.x; y/.x/ dx: ˝

Note that when, in the deﬁnition above, ˝ D 3 we will denote the operators P and R by P and R, respectively The following theorem presents the relationships between the parametrix-based volume and surface potentials and their counterparts for the Laplace equation (a Á 1). It is now rather simple to obtain, similar to [CMN13], the mapping properties, jump relations, and invertibility results for the parametrix-based surface and volume potentials. The following relations coincide with their analogous relations for the potentials in bounded domains which appear in [MiPo15b]. Theorem 1 The operators V; W; V; W; W0, and L satisfy the following relations for their counterparts associated with the Laplace operator: Á P D P ; R DrŒP.r ln a/; a Á Á V D V ; V D V ; a a Â Ã Â Ã @ ln a @ ln a W D W V ; W D W V ; @n @n Â Ã Á @ 0 0 ˙ ˙ ln a W D aW ; LO WD aL; L D LO a W : a @n

Remark 1 The subscript refers to the analogous surface potentials with a Á 1, i.e. PjaD1 D P. Note that P is the fundamental solution of the Laplace equation. C Furthermore, in virtue of the Lyapunov-Tauber theorem L D L D L. One of the main differences with respect to the bounded domain case is that the integrands of the operators V, W, P, and R and their corresponding direct values and conormal derivatives do not always belong to L1. In these cases, the integrals should be understood as the corresponding duality forms (or the limits of these forms for the inﬁnitely smooth functions, existing due to their density in corresponding Sobolev spaces). 19 A New Family of BDIEs 221

Condition 2 In addition to conditions (19.3) and (19.11), we will also sometimes assume the following condition:

!2a 2 L1.˝/: (19.12)

Remark 2 Note as well that due to the essential boundedness of the function a and the continuity of the function ln a, the components of r ln a and ln a will be essentially bounded as well. Theorem 2 The following operators are continuous under condition (19.11):

1 1 V W H 2 .S/ ! H .˝/;

1 1 W W H 2 .S/ ! H .˝/:

Corollary 1 The following operators are continuous under condition (19.11) and (19.12),

1 1;0 V W H 2 .S/ ! H .˝I A/;

1 1;0 W W H 2 .S/ ! H .˝I A/:

Theorem 3 The following operators are continuous under condition (19.11),

P W H 1.3/ ! H 1.3/; R W L2.!1I 3/ ! H 1.3/; P W He 1.˝/ ! H 1.3/:

Theorem 4 The following operators are continuous under condition (19.11) and (19.12),

P W L2.!I ˝/ ! H 1;0.3I A/; R W H 1.˝/ ! H 1;0.˝I A/:

19.6 Third Green Identities and Integral Relations

Let B.y/ be the ball centred at y 2 ˝ with radius sufﬁciently small. Then, 2 1;0 R.; y/ 2 L .!I ˝ XB.y// and thus P.; y/ 2 H .˝ XB.y//. Applying the second 1;0 Green identity (19.6) with v D P.; y/ and any u 2 H .˝I A/ in ˝ X B.y/ and using standard limiting procedures as ! 0 (cf. [Mr70]) we obtain the third Green identity (integral representation formula) for the function u 2 H 1;0.˝I A/: 222 S.E. Mikhailov and C.F. Portillo

u C Ru VTCu C W Cu D PAu; in ˝: (19.13)

If u 2 H1;0.˝I A/ is a solution of the partial differential equation (19.7), then, from (19.13), we obtain

u C Ru VTC.u/ C W Cu D Pf ; in ˝: (19.14)

Taking the trace of (19.14), we obtain 1 Cu CRu VTCu W Cu CPf ; on S: 2 C C D (19.15)

For some distributions f ; , and , we consider an indirect integral relation associated with the third Green identity (19.14)

u C Ru V C W˚ D Pf ; in ˝: (19.16)

Appropriately modifying the proofs of Lemma 4.1 in [CMN13] and Lemma 9.4.1 in [MiPo15b], one can prove the following assertion. 1 1 1 Lemma 1 Let u 2 H .˝/,f 2 L2.!I ˝/, 2 H 2 .S/ and ˚ 2 H 2 .S/ satisfy relation (19.16). Let conditions (19.11) and (19.12) hold. Then u 2 H 1;0.˝; A/, solves the equation Au D fin˝ and the following identity is satisﬁed

V. TCu/ W.˚ Cu/ D 0 in ˝:

The following statement is the counterpart of Lemma 4.3 in [MiPo15b]for exterior domains. The proof follows from the invertibility of the operator V (see Corollary 8.13 in [McL00]). 1 Lemma 2 Let 2 H 2 .S/.If

V .y/ D 0; y 2 ˝; then D 0.

19.7 Boundary-Domain Integral Equation System

1 1 Let the functions ˚0 2 H 2 .S/ and 0 2 H 2 .S/ be continuous ﬁxed extensions 1 1 e 1 to S of the functions 0 2 H 2 .SD/ and 0 2 H 2 .SN /. Moreover, let 2 H 2 .SN / e 1 and 2 H 2 .SD/ be arbitrary functions formally segregated from u (cf. [CMN09, CMN13, MiPo15b]. 19 A New Family of BDIEs 223

We will derive a system of BDIEs for the BVP (19.7)–(19.9) substituting the functions

C C u D ˚0 C ; T u D 0 C ; on SI (19.17) in the third Green identities (19.14) and (19.15). In what follows, we will denote by X the vector of unknown functions

1 1 > 1;0 e 2 e 2 X D .u; ; / 2 WD H .˝I A/ H .SD/ H .SN / where

1 1 1 e 2 e 2 WD H .˝/ H .SD/ H .SN /:

We substitute the functions (19.17)in(19.14) and (19.15) to obtain the following BDIE system (M12)

u C Ru V C W D F0 in ˝; (19.18a) 1 C C Ru V W F0 ˚0 on S; 2 C C D (19.18b) where F0 D Pf C V 0 W˚0. We denote by M12 the matrix operator that deﬁnes the system .M12/: 2 3 I C R VW M12 D 4 1 5 ; CR V I W 2 C and by F 12 the right-hand side of the system:

12 C > F D Œ F0; F0 ˚0 :

The system (M12) can be expressed in terms of the matrix notations as

M12X D F 12:

If the conditions (19.11) and (19.12) hold, then due to the mapping properties of the potentials, F 12 2 ¿12 12, while operators M12 W ! ¿12 and M12 W ! 12 are continuous. Here, we denote

12 1;0 1 ¿ WD H .˝; A/ H 2 .S/;

12 1 1 WD H .˝/ H 2 .S/: 224 S.E. Mikhailov and C.F. Portillo

A proof of the following assertion is similar to the proof of the corresponding Theorem 9.5.1 for the interior domain in [MiPo15b]. 1 Theorem 5 (Equivalence) Let f 2 L2.!I ˝/,let˚0 2 H 2 .S/ and let 0 2 1 1 1 H 2 .S/ be some ﬁxed extensions of 0 2 H 2 .SD/ and 0 2 H 2 .SN /, respectively. Let conditions (19.11) and (19.12) hold. i) If some u 2 H 1;0.˝I A/ solves the BVP (19.7)–(19.9), then the triplet > 1;0 e 1 e 1 .u; ; / 2 H .˝I A/ H 2 .SD/ H 2 .SN / where

C C D u ˚0; D T u 0; on S;

solves the BDIE system (M12). > 1;0 e 1 e 1 ii) If a triple .u; ; / 2 H .˝I A/H 2 .SD/H 2 .SN / solves the BDIE system (M12), then this solution is unique. Furthermore, u solves the BVP (19.7)–(19.9) and the functions ; satisfy

C C D u ˚0; D T u 0; on S:

19.8 Invertibility

In this section, we aim to prove the invertibility of the operator M12 W ! ¿12 by showing ﬁrst that the arbitrary right-hand side ¿12 from the respective spaces can be represented in terms of the parametrix-based potentials and using then the equivalence theorems. The following result can be proved similar to its counterpart, Corollary 7.1 in [CMN13] with another parametrix. The analogous result for bounded domains can be found in Lemma 3.5 in [CMN09]. Lemma 3 Let

1;0 1 .F0; F1/ 2 H .˝I A/ H 2 .@˝/:

Then there exists a unique triplet .f; ;˚/ such that .f; ;˚/ D > 1;0 1 2 1 1 C.F0; F1/ , where C W H .˝; A/ H 2 .S/ ! L .!I ˝/ H 2 .S/ H 2 .S/ is a linear bounded operator and .F0; F1/ are given by

F0 D Pf C V W˚ in ˝ C F1 D F0 ˚ on @˝

Employing Lemma 3 and the arguments as in the proof of Theorem 7.1 in [CMN13], it is possible to prove one of the main results on the invertibility of the matrix operator of the BDIE system (M12). 19 A New Family of BDIEs 225

Theorem 6 If conditions (19.11) and (19.12) hold, then the following operator is continuous and continuously invertible:

M12 W ! ¿12:

Let us introduce the additional condition

lim !.x/ra.x/ D 0: (19.19) x!1

Now, in a similar fashion as in Lemma 7.2 in [CMN13], we we can prove the following assertion. Lemma 4 Let conditions (19.11) and (19.19) hold. Then, for any >0the operator R can be represented as R D Rs C Rc, where k Rs kH1.˝/<, while 1 1 Rc W H .˝/ ! H .˝/ is compact. Since the limit of a converging sequence of compact operators is also compact, Lemma 4 implies the following result. Corollary 2 Let conditions (19.11) and (19.19) hold. Then the operator R W H 1.˝/ ! H 1.˝/ is compact and the operator I C R W H 1.˝/ ! H 1.˝/ is Fredholm with zero index. Theorem 7 If conditions (19.11),(19.12), and (19.19) hold, then the operator

M12 W ! 12; is continuously invertible. The theorem can be proved by implementing Corollary 2, Fredholm alternative and Theorem 5; cf. Theorem 7.4 in [CMN13].

References

[CMN09] Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefﬁcient, I: Equivalence and invertibility. J. Integr. Equ. Appl. 21, 499–543 (2009) [CMN13] Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct segregated boundary-domain integral equations for variable-coefﬁcient mixed BVPs in exterior domains. Anal. Appl. 11(4), 1350006 (2013) [Co88] Costabel, M.: Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal. 19, 613–626 (1988) [Gr78] Giroire, J., Nedelec, J.: Numerical solution of an exterior Neumann problem using a double layer potential. Math. Comput. 32, 973–990 (1978) [Ha71] Hanouzet, B.: Espaces de Sobolev avec Poids. Application au Probleme de Dirichlet Dans un Demi Espace. Rendiconti del Seminario Matematico della Universita di Padova 46, 227–272 (1971) 226 S.E. Mikhailov and C.F. Portillo

[McL00] McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000) [Mi02] Mikhailov, S.E.: Localized boundary-domain integral formulations for problems with variable coefﬁcients. Eng. Anal. Bound. Elem. 26, 681–690 (2002) [Mi11] Mikhailov, S.E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378, 324–342 (2011) [MiPo15a] Mikhailov, S.E., Portillo, C.F.: BDIE system to the mixed BVP for the Stokes equations with variable viscosity. In: Constanda, C., Kirsh, A. (eds.) Integral Methods in Science and Engineering: Theoretical and Computational Advances. Springer (Birkhäuser), Boston (2015) [MiPo15b] Mikhailov, S.E., Portillo, C.F.: A new family of boundary-domain integral equations for a mixed elliptic BVP with variable coefﬁcient. In: Proceedings of the 10th UK Conference on Boundary Integral Methods. Brighton University Press, Brighton (2015) [Mr70] Miranda, C.: Partial Differential Equations of Elliptic Type, 2nd edn. Springer, Berlin (1970) Chapter 20 Radiation Conditions and Integral Representations for Clifford Algebra-Valued Null-Solutions of the Iterated Helmholtz Operator

D. Mitrea and N. Okamoto

20.1 Introduction

Throughout this chapter fix n 2 , n 2, and setbx WD x=jxj for x 2 n nf0g.Call ˝ Â n an exterior domain if its complement n n ˝ is a compact set. Our main goal is to develop integral representations for Clifford-valued null-solutions of the iterated Helmholtz operator in exterior domains with Lipschitz boundary. The layout of this chapter is as follows. In Section 20.1 we briefly review the Clifford algebra setting necessary for our construction, allowing us to factor 2 the Helmholtz operator C k via the perturbed Dirac operator Dk. Then in Section 20.2 we develop fundamental solutions of the iterated Helmholtz and iterated perturbed Dirac operators respectively, exploring both their singular and asymptotic behavior along the way. Finally in Section 20.3 we use integration by parts (see Theorem 1 below) to generate integral representations for null-solutions of the iterated Helmholtz operator, first over bounded Lipschitz domains, and then (with the addition of appropriate radiation conditions) we extend our result to Lipschitz exterior domains. Given some number m 2 0, the (complex) Clifford algebra .C`m; C; ˇ/ is the minimal enlargement of ¼m to a unitary complex algebra, which is not generated (as an algebra) by any proper subspace of ¼m, and such that

2 m m x ˇ x Djxj for any x 2 ,! ¼ ,! C`m:

This identity readily implies that, if fejg1ÄjÄm is the standard orthonormal basis in m , then ej ˇ ej D1 and ej ˇ ek Dek ˇ ej whenever 1 Ä j ¤ k Ä m.

D. Mitrea () • N. Okamoto University of Missouri, Columbia, MO, USA e-mail: [email protected]; [email protected]

Any element u 2 C`m can be uniquely represented in the form

m X X X 0 u D uI eI D uI eI; uI 2 ¼: (20.1) I `D0 jIjD`

. ; ;:::; / 1 Here eI stands for the product ei1Pˇ ei2 ˇˇei` if I D i1 i2 i` and e; WD is the multiplicative unit. Also, 0 indicates that the sum is performed only over strictly increasing multi-indices I D .i1; i2;:::;i`/ with 1 Ä i1 < i2 < < i` Ä m. Subsequently, we will work with Clifford algebra-valued functions u as in (20.1) which have their coefficients uI belonging to a certain Banach space. Specifically, given an open set ˝ Â n, we will work with u as in (20.1) with the additional 1 1 property that all uI’s belong to C .˝/, in which case we will write u 2 C .˝; C`m/. 0 1 Similarly, we will use the notation u 2 C .˝;C`m/,oru 2 L .˝; C`m/,oru 2 1 .˝; C` / ˝ Lloc m , etc., whenever the coefficients uI are continuous on the closure of , or are absolutely integrable on ˝, or are absolutely integrable on compact subsets of ˝,etc.

When simultaneously dealing with two Clifford algebras, say C`m1 and C`m2 ,we canonically view them as the subalgebras of C`m where m WD maxfm1; m2g freely generated by fe1;:::;em1 g and fe1;:::;em2 g, respectively. Here is a concrete case of interest where this convention is called for. Let ˝ be an open set in n. Then the classical (homogeneous) Dirac operator associated with n is given by

Xn D WD ej ˇ @j: (20.2) jD1

1 This acts on a function u 2 C .˝; C`m/ where m 2 0 according to

Xn Du WD ej ˇ .@ju/ (20.3) jD1 with the right-hand side of (20.3) regarded as a C`M-valued function, where M WD maxfn; mg. We shall also work with the perturbed Dirac operator

Dk WD D C kenC1; (20.4) for some complex number k 2 ¼. Hence, given m 2 0 and an arbitrary u 2 1 C .˝; C`m/,wehave

Xn Dku D ej ˇ @ju C kenC1 ˇ u; (20.5) jD1 20 Integral Representations for the Iterated Helmholtz Operator 229 with the right-hand side of (20.5) regarded as a C`M-valued function in ˝,forM WD maxfn C 1; mg. When the Dirac operator D and the perturbed Dirac operator Dk are 1 acting from the right on some u 2 C .˝; C`m/ we write uD and uDk, respectively. Hence, in this scenario, Xn Xn uD D @ju ˇ ej; uDk D @ju ˇ ej C ku ˇ enC1: (20.6) jD1 jD1

One of the basic properties of the Dirac operators introduced above is that they can be thought of as square-roots of familiar second-order differential operators. More precisely, D and Dk satisfy 2 2 . 2/; D D and Dk D C k (20.7) P n @2 n where WD jD1 j is the usual Laplace operator in . We conclude this section with an integration by parts formula involving perturbed Dirac operators (for a proof see [ON16]). In the sequel, the .n 1/-dimensional Hausdorff measure in n will be denoted by H n1. Theorem 1 Let ˝ be an open subset of n with compact Lipschitz boundary and denote its outward unit normal by D . 1;:::; n/. Let k 2 ¼ be arbitrary. Suppose 0 u;v 2 C .˝;C`m/ satisfy

. /; . v/ 1 .˝; C` / . / v . v/ 1.˝; C` /: uD D 2 Lloc m and uDk ˇ C u ˇ Dk 2 L m

If ˝ is unbounded, also assume that Z x lim u.x/ ˇ ˇ v.x/ dH n1.x/ D 0: R!1 jxjDR jxj Then the following integration by parts formula holds: Z n o .uDk/.x/ ˇ v.x/ C u.x/ ˇ .Dkv/.x/ dx ˝ Z D u.x/ ˇ .x/ ˇ v.x/ dH n1.x/: @˝

20.2 Fundamental Solutions

.1/ Denote by H ./ the Hankel function of the ﬁrst kind with index 2 (cf. e.g., [DLMF10]), and deﬁne the constant 1 c WD : n 4i.2 /.n2/=2 230 D. Mitrea and N. Okamoto

Seeking a fundamental solution of the iterated Helmholtz operator .Ck2/N , where N 2 , deﬁne

. 1/N1 .n2N/=2 .1/ . / . / cnk H. 2 /=2 kjxj ˚ N .x/ WD n N ; 8 x 2 n nf0g: (20.8) k 2N1.N 1/Š jxj.n2N/=2

1 ˚ .1/ Corresponding to N D , we have that k is a fundamental solution of the Helmholtz operator C k2 in n (for a proof see [MMMM]). It is well known .1/ that for each 2 , the Hankel function H ./ is analytic on ¼ n .1;0, i.e. the complex plane with a branch cut along the non-positive real axis, so ˚ .N/ C 1.n 0/ k 2 n for every N 2 . As such, via a direct computation we obtain the following lemma. Lemma 1 For any N 2 ,k2 .0; 1/, we have the following recursive property:

. 2/˚ .NC1/. / ˚ .N/. / n 0 : C k k x D k x pointwise at every x 2 nf g

The next two lemmas involve the Dirac and the perturbed Dirac operators (20.2) ˚ .N/ and (20.4) acting on k . Lemma 2 For any N 2 ,k2 .0; 1/, the following recursive property holds:

.N/ . 1/ ˚ .x/ D˚ NC .x/ D k x; 8 x 2 n nf0g: k 2N Moreover, .1/ . / .1/ H =2 kjxj D˚ .x/ Dc kn=2 n x; 8 x 2 n nf0g: k n jxjn=2

˚ .N/ This leads us to the following explicit formula for Dk k , which plays an important role in the integral representation (20.13) we develop in Section 20.3. Lemma 3 For any N 2 ,k2 .0; 1/, we have

N .n2NC2/=2 h i .N/ .1/ cnk .1/ .1/ Dk˚ .x/ D H .kjxj/bx H .kjxj/ enC1 k 2N1 .N 1/Š jxj.n2N/=2 .n2NC2/=2 .n2N/=2 pointwise at every x 2 n nf0g. A close inspection of (20.8) combined with the behavior of the Hankel function near the origin yields the following lemma. Lemma 4 Let N 2 and k 2 .0; 1/. Then

ˇ . / ˇ ˇ˚ N .x/ˇ lim k D 1; x!0 F.x/ 20 Integral Representations for the Iterated Helmholtz Operator 231 where 8 ˆ ..n 2N/=2/ 1 n ˆ if N < ; ˆ22N n=2 . 1/Š n2N 2 ˆ N jxj <ˆ 1 ˇ ˇ n F.x/ D ˇ ln.kjxj/ˇ if N D ; 8 x 2 n nf0g; ˆ2n1 n=2 . 1/Š 2 ˆ N ˆ ˆ ..2N n/=2/ n :ˆ if N > : 2n n=2 .N 1/Š k2Nn 2

Consequently, for each R 2 .0; 1/ there exists a ﬁnite constant CD C.R; n; N; k/>0 with the property that 8 1 ˆ 1 < n; ˆC C 2 if N ˆ jxjn N 2 ˇ ˇ < ˇ ˇ ˇ .N/ ˇ n ˚ .x/ Ä C 1 C ˇ ln.jxj/ˇ if N D ; k ˆ 2 ˆ ˆ n :C N > ; if 2

n 0< ˚ .N/ n for every x 2 satisfying jxjÄR. Thus k is locally integrable in . ˚ .N/ Relying on Lemmas 1–4, we prove that k is a fundamental solution of . C k2/N in n, for each N 2 . Theorem 2 Suppose n 2 ,n 2, and fix k 2 .0; 1/. Then for any N 2 ,the ˚ .N/ locally integrable function k defined in (20.8) is a fundamental solution of the iterated Helmholtz operator . C k2/N in n. ˚ .N/ ˚ .N/ Next we analyze the behavior of k and Dk k at infinity. To simplify notation, we introduce the constants

.n2N1/=2 i .nC2N1/=4 . / k e b N WD for each N 2 : n;k 2.nC2N1/=2 .n1/=2 .N 1/Š

Lemma 5 Fix N 2 ,k2 .0; 1/. Then

ikjxj ikhbx;yi . / . / e e ˚ N .x y/ D b N C O jxj.n2NC3/=2 k n;k jxj.n2NC1/=2 as jxj!1uniformly for y in compact subsets of n, and

ikjyj ikhxb;y i . / . / e e ˚ N .x y/ D b N C O jyj.n2NC3/=2 k n;k jyj.n2NC1/=2 as jyj!1uniformly for x in compact subsets of n. 232 D. Mitrea and N. Okamoto

Lemma 6 Fix N 2 ,k2 .0; 1/. Then ˚ .N/ . / ˚ .N/. /. b / .n2NC3/=2 Dk k x y D k x y ik x C kenC1 C O jxj

ikjxj ikhbx;yi .N/ e e .n2NC3/=2 D kb .ibx C e 1/ C O jxj n;k jxj.n2NC1/=2 nC as jxj!1uniformly for y in compact subsets of n, and ˚ .N/ . / ˚ .N/. /. b / .n2NC3/=2 Dk k x y D k x y ik y C kenC1 C O jyj

ikjyj ikhxb;y i .N/ e e .n2NC3/=2 D kb .iby C e 1/ C O jyj n;k jyj.n2NC1/=2 nC as jyj!1uniformly for x in compact subsets of n. ˚ .N/ Since k is a fundamental solution of the iterated Helmholtz operator, and the perturbed Dirac operator provides a “square root” of the Helmholtz operator (see (20.7)), we are motivated to deﬁne the C`nC1-valued function 8 <ˆ. 1/N=2 ˚ .N=2/. / k x if N is even, .N/. / n 0 : k x WD ˆ for all x 2 nf g :. 1/.NC1/=2 ˚ ..NC1/=2/. / Dk k x if N is odd, (20.9) .N/ Note that an explicit formula for k in terms of Hankel functions may be readily obtained by combining this deﬁnition with (20.8) and Lemma 3. Lemma 7 For any N 2 ,k2 .0; 1/, we have the following recursive property:

.NC1/. / .N/. /; n 0 : Dk k x D k x 8x 2 nf g

.N/ Concerning the behavior of k near the origin, we have the following result. Lemma 8 Fix N 2 and k 2 .0; 1/. Then 8 ˆO x Nn N < n; ˆ j j if <ˆ .N/ .x/ D O ln jxj if N D n; as x ! 0: k ˆ ˆ :ˆ O 1 if N > n; ˇ ˇ ˇ.n/. /ˇ .1/ Corresponding to N D n and n being odd, the stronger result k x D O as 0 .N/ n x ! holds. In particular, k is locally integrable in . 20 Integral Representations for the Iterated Helmholtz Operator 233

.N/ To analyze the behavior of k at infinity, apply Lemma 5 to (20.9) when N is even, and apply Lemma 6 to (20.9) when N is odd. Theorem 3 Suppose n 2 ,n 2, and fix k 2 .0; 1/. Then for any N 2 , C` .N/ the locally integrable nC1-valued function k defined in (20.9) is a fundamental N n solution of the perturbed Dirac operator Dk in .

20.3 Integral Representations

We are ready to discuss our integral representations for null-solutions of the iterated Helmholtz operator. Taking advantage of the fact that the action of the perturbed Dirac operator from the right (recall (20.6)) and from the left (recall (20.3)) is the .N/ .N/ .N/ n 0 same on k (i.e. k Dk Á Dk k on nf g), the ﬁrst lemma in this section is an application of the integration by parts formula in Theorem 1. Lemma 9 Fix N 2 ,k 2 .0; 1/, and suppose ˝ is an open bounded subset of n with Lipschitz boundary and outward unit normal . Consider an arbitrary N C`m-valued function u 2 C .˝;C`m/. Then for each M 2f1;2;:::;Ng and every x 2 ˝, one has Z M.N/ . / . / Dk k y x ˇ u y dy ˝nB.x;"/ Z MX1 . 1/j .NMCjC1/. / . / j . / H n1. / D k y x ˇ y ˇ Dku y d y @˝ jD0

MX1 Z . 1/ y x . 1/j NMCjC . / j . / H n1. / k y x ˇ ˇ Dku y d y @ . ;"/ " jD0 B x Z . 1/M .N/. / M . / C k y x ˇ Dku y dy ˝nB.x;"/ " 0; 1 . ;@˝/ for every 2 2 dist x . An important consequence of Lemma 9 corresponds to the case when M is chosen to be equal to N and N is an even number. Lemma 10 Fix N 2 ,k2 .0; 1/, and suppose ˝ is an open bounded subset of n with Lipschitz boundary and outward unit normal . Consider the C`m-valued function

2N 2 N u 2 C .˝;C`m/ satisfying . C k / u D 0 in ˝: 234 D. Mitrea and N. Okamoto

Then for every x 2 ˝, Z 2XN1 . / . 1/j .jC1/. / . / j . / H n1. / u x D k y x ˇ y ˇ Dku y d y @˝ jD0 Z XN n ˚ .j/ . / . / . 2/j1 . / D Dk k x y ˇ y ˇ C k u y @˝ jD1 o ˚ .j/. / . / . 2/j1 . / H n1. /: C k x y y ˇ Dk C k u y d y (20.10)

Aiming to extend this result to an exterior domain ˝, consider B.0; R/ (the ball centered at the origin with radius R >0), where R is large enough so that n n˝ DW ˝c B.0; R/. Applying (20.10)to˝ \ B.0; R/ gives us an integral representation that includes a boundary integral over @B.0; R/. In the main result presented below, appropriate radiation conditions are imposed on u to insure that the integral over @B.0; R/ will vanish as R !1. Theorem 4 Fix N 2 ,k2 .0; 1/, and let ˝ be an exterior domain in n with Lipschitz boundary and outward unit normal . Consider the C`m-valued function

2N 2 N u 2 C .˝;C`m/ satisfying . C k / u D 0 in ˝:

Suppose u satisﬁes the radiation condition Z ˇ ˇ2 ˇ ˇ n1 ˇikwN .y/ Cby ˇ DwN .y/ˇ dH .y/ D o.1/ as R !1; (20.11) jyjDR where

1 XN jyjj w .y/ WD . C k2/j u.y/; 8 y 2 ˝: N jŠ.2ik/j jD0

Assume also that u satisﬁes the damping conditions 8 Z ˇ ˇ <ˆ . 2j/ ˇ ˇ2 o R if j is even, ˇDj u.y/ˇ dH n1.y/ D (20.12) k :ˆ jyjDR o.R3j/ if j is odd, 20 Integral Representations for the Iterated Helmholtz Operator 235 as R !1; 8j 2f0; 1; : : : ; 2N 1g. Then for every x 2 ˝ Z XN h ˚ .j/ . / b . 2/j1 . / Dk k x y ˇ y ˇ C k u y jyjDR jD1 i ˚ .j/. /b . 2/j1 . / H n1. / C k x y y ˇ Dk C k u y d y

D o.1/ as R !1:

Consequently, for every x 2 ˝ Z XN n . / ˚ .j/ . / . / . 2/j1 . / u x D Dk k x y ˇ y ˇ C k u y @˝ jD1 o ˚ .j/. / . / . 2/j1 . / H n1. /: C k x y y ˇ Dk C k u y d y (20.13)

Notice that when N D 1,(20.11) becomes Rellich’s adaption of the celebrated Sommerfeld radiation condition (cf. [Sc92] and the references therein). As the Sommerfeld radiation condition was originally accompanied by a so-called “Som- merfeld ﬁniteness condition” later shown to be superﬂuous, when N D 1 it turns out that (20.11) implies (20.12)(cf.[MMMM]), so explicitly stating our damping conditions in this case is unnecessary. Whether that implication continues to hold for N >1is still an open question.

Acknowledgements The ﬁrst author has been supported in part by a Simons Foundation grant # 426669.

References

[MMMM] Marmolejo-Olea, E., Mitrea, D., Mitrea, I., Mitrea, M.: Radiation Conditions and Integral Representations for Clifford Algebra-Valued Null-Solutions of the Helmholtz Operator, preprint (2016) [ON16] Okamoto, N.: Radiation conditions and integral representations for Clifford algebra- valued null-solutions of the iterated perturbed Dirac operator. Ph.D. Thesis, University of Missouri (2017) [DLMF10] Olver, F., Maximon, L.: Bessel functions. In: Olver, F., Lozier, D., Boisvert, R., Clark, C. (eds.) NIST Handbook of Mathematical Functions, pp. 217–223. Cambridge University Press, New York, NY (2010) [Sc92] Schot, S.: Eighty years of Sommerfeld’s radiation condition. Hist. Math. 19, 393–396 (1992) Chapter 21 A Wiener-Hopf System of Equations in the Steady-State Propagation of a Rectilinear Crack in an Inﬁnite Elastic Plate

A. Nobili, E. Radi, and L. Lanzoni

21.1 Introduction

Let us consider a Kirchhoff–Love inﬁnite plate with thickness h, supported by a Winkler elastic foundation (Figure 21.1). The plate has a linear crack and a Cartesian reference frame is introduced such that the crack corresponds to the negative part of the x1-axis. Besides, the crack is linearly propagating at a constant speed v (steady- state propagation) and the origin of the Cartesian reference frame is attached to and moves along with the crack tip. The governing equation for the transverse displacement of the plate w reads

D44w C kw Dh@ttw C q; (21.1)

being 4D@x1x1 C @x2x2 the Laplace operator in two dimensions, q the transverse distributed load per unit area, D the plate bending stiffness, k the Winkler modulus, and the mass density per unit volume [No14]. We let w D w.x1 vt; x2/ and Equation (21.1) may be rewritten as q 44w C 2@ w C 4w D ; (21.2) x1x1 D

A. Nobili ()•E.Radi University of Modena and Reggio Emilia, Modena, Italy e-mail: [email protected]; [email protected] L. Lanzoni University of San Marino, San Marino, San Marino e-mail: [email protected]

Fig. 21.1 Cracked Kirchhoff plate resting on a Winkler elastic foundation. The crack is propagating in the x direction with speed v

having let the characteristic lengths r s 4 D D D ; and D ; k hv2 together with the positive dimensionless ratio

D =:

1 We rescale the co-ordinate axes .x; y/ D .x1; x2/ and take q Á 0, with no loss of generality. Then, Equation (21.2) becomes

2 4O 4O w C @xxw C w D 0; (21.3) where 4DO @xx C @yy is the Laplacian operator in the dimensionless co-ordinates x; y. The special case D 0 corresponds to the static problem, whose solution is given in [AnEtAl63] and extended in [NoEtAl15]. The Fourier transform of w.x; y/ along x is deﬁned on the real axis in the usual way Z C1 F Œw.s; y/ DNw.s; y/ w.x; y/ exp.isx/dx 1 21 Steady-State Crack Propagation 239 along with the inverse transform Z 1 C1 F 1ŒwN .x; y/ D w.x; y/ wN .s; y/ exp.isx/ds: 2 1

In the same fashion, the unilateral (or generalized) transforms are introduced: Z C1 F CŒw.s; y/ DNwC.s; y/ w.x; y/ exp.{sx/dx 0 which is analytic in the complex half-plane Im.s/>˛1 provided that 9 ˛1 >0 such that w.x; y/

Z 0 F Œw.s; y/ DNw.s; y/ w.x; y/ exp.{sx/dx; 1 which is analytic in the complex half-plane Im.s/<˛2 provided that 9 ˛2 >0such that w.x; y/>W2.y/ exp.˛2x/ as x !1. Consequently, the bilateral Fourier transform is related to the unilateral transforms through

wN .s; y/ DNwC.s; y/ CNw.s; y/ (21.4) and it is analytic in the inﬁnite strip S Dfs 2 ¼ W˛1 < Im.s/<˛2g containing the real axis. Taking the Fourier transform of Equation (21.3)inthex variable, a linear constant coefﬁcient ODE is obtained whose general solution is p p wN .s; y/ D A1 exp. 1jyj/ C B1 exp. 1jyj/ p p C A2 exp. 2jyj/ C B2 exp. 2jyj/; wherein p 2 2 2 1;2 D s s 1; (21.5)

4 2 2 such that 12 D s s C 1. Hereinafter, Re.s/ and Im.s/ denote the real and the imaginary part of s 2 ¼, respectively, and a superscript asterisk denotes complex conjugation, i.e. s D Re.s/ { Im.s/.Letw.x; yC/ (w.x; y/) be the restriction of the displacement w.x; y/ in the upper (lower) half of the .x; y/-plane, respectively. It is understood that yC 2 .0; C1/ and y 2 .1;0/. The general solution of the ODE (21.3), bounded at inﬁnity, is p p ˙ ˙ ˙ wN .s; y / D A1 exp. 1jyj/ C A2 exp. 2jyj/; (21.6) 240 A. Nobili et al.

˙ ˙ where A1 ; A2 are four complex-valued functions of s to be determined. Thep square root in (21.6) is made deﬁned by choosing the Riemann sheet such that Re. 1;2/> 0. Besides, we let the shorthand notation for the restriction of (21.6)tothex-axis

˙ ˙ wN 0˙ .s/ D A1 C A2 (21.7)

˙ and let Ai split into symmetric and skew-symmetric parts ˙ 1 N ; 1; 2: Ai D 2 Ai ˙ Ai i D (21.8)

21.2 Boundary Conditions

Let the bending moment and equivalent shearing force (deprived of the factor D) m D @yy C @xx w;vD@y @yy C .2 /@xx w; together with the slope

D @yw:

The boundary conditions (BCs) across the line y D 0 ahead of the crack tip are of the kinematic type as they warrant continuity for displacement and slope

w0.x/ D 0.x/ D 0; x >0; (21.9) and of the static type, for they demand continuity for bending moment and equivalent shearing force

m0.x/ D v0.x/ D 0; x >0: (21.10)

Here, f .0/ denotes the jump of the function f .y/ across y D 0, namely f .0C/ f .0/, while a subscript zero stands for evaluation on the real axis. Conversely, it is assumed that the crack ﬂanks are loaded in a continuous fashion by a harmonic loading. Then, the BCs at the crack line y D 0 are

˙ ˙ m.x;0 / D M0 exp.{ax/; v.x;0 / D V0 exp.{ax/; x <0; (21.11) where M0 D M0.a/, V0 D V0.a/ and

Im.a/<0 (21.12) to ensure a decay condition as x !1. We observe that the system (21.11), just like the system (21.9,21.10), entails four conditions, for it applies at both ﬂanks of the crack (denoted by y D 0˙). As a consequence of loading continuity, 21 Steady-State Crack Propagation 241

Equation (21.10) hold on the entire real line. Furthermore, we note that a similar problem arises in the realm of couple-stress theory, although with a different set of boundary conditions [Ra08, MiEtAl12, PiEtAl12]. Equation (21.9) may be written in terms of the generalized plus transform as

C C wN 0 D N0 D 0; whence, using the general solution (21.6) and in view of Equations (21.4, 21.8), it is

A1 C A2 D wN 0 ; (21.13a) p p 1AN 1 2AN 2 D N0 : (21.13b)

Likewise, the Fourier unilateral transform of Equation (21.11)gives

M0 V0 mN 0 D{ ; vN0 D{ ; s C a s C a in the strip S1 Dfs W Im.s/< Im.a/g. In particular, inequality (21.12) guarantees the existence of the minus transform up to a little above the real axis in the lower complex half-plane. Thus 2 ˙ 2 ˙ C M0 .1 s /A1 C .2 s /A2 DNm { ; 0 s C a (21.14a) n p p o 2 ˙ 2 ˙ C V0 ˙ 1 1 .2 /s A1 C 2 2 .2 /s A2 DNv { ; 0 s C a (21.14b) in the strip S0 D S \ S1. Finally, the Fourier transforming Equation (21.10), which are holding on the entire real line, gives

2 2 .1 s /A1 C .2 s /A2 D 0; (21.15a) p p 2 2 1 1 .2 /s AN 1 C 2 2 .2 /s AN 2 D 0; (21.15b) according to which the system (21.14) becomes 1 2 2 C M0 .1 s /AN 1 C .2 s /AN 2 DNm { ; 2 0 s C a (21.16a) n p p o 1 2 2 C V0 1 1 .2 /s A1 C 2 2 .2 /s A2 DNv { : 2 y0 s C a (21.16b) 242 A. Nobili et al.

Conditions (21.15) are immediately fulﬁlled through letting

2 A1 D.2 s /A; 2 A2 D .1 s /A; p 2 AN 1 D 2 2 .2 /s AN ; p 2 AN 2 D 1 1 .2 /s AN :

Then, Equation (21.13) become

.1 2/ A D wN 0 ; (21.18a) p 12 .1 2/ AN D N0 : (21.18b)

Likewise, the system (21.16)gives

C M0 .2 1/ K.s/AN DNm { ; (21.19a) 0 s C a

C V0 .2 1/ K.s/A DNv { ; (21.19b) 0 s C a where the kernel function K.s/ is let as follows: p 2 2 2 .2 1/ K.s/ D 1 1 .2 /s .2 s / p 2 2 C 2 2 .2 /s .1 s /: (21.20)

With the help of Equation (21.5), Equation (21.20) may be rewritten as q p p p Á2 2 2 2 2 2 2 2 2 4 s 1K.s/ D s s 1 0s C s 1 q p p Á2 2 2 2 2 2 2 s C s 1 0s s 1 ; (21.21) p 2 2 having let 0 D 1 . In particular, in the limit as ! 0 and with s 1 ! {, the kernel 4{K.s/ in Equation (21.21) reduces to Eq.(24) of [AnEtAl63]. Solving the system (21.18) for the unknown functions AN ;A and plugging the result in Equation (21.19) provides the following two uncoupled Wiener-Hopf (W-H) equations, namely

C V0 K.s/wN 0 CNv D { ; 0 s C a

1=2 C M0 .12/ K.s/ N0 Nm D{ : 0 s C a 21 Steady-State Crack Propagation 243

Making use of Equation (21.5), this system of functional equations may be rewritten as

C V0 K.s/wN 0 CNv D { ; (21.22a) 0 s C a . / K s C M0 p q N0 Nm0 D{ ; (21.22b) 2 s C a s2 ˇ2 s2 ˇ

2 2 2 2 having taken the factor decomposition 12 D .s ˇ /.s ˇ /, with r q ˇ 1 2 1 2 2 1: D 2 C 2

ˇ It is observed that is a complex number with unit modulusp and it is located in the 0 < 2 ﬁrst quadrantp of the complex plane inasmuch as Ä . Alternatively, when 2, we deﬁne the positive real numbers ˇ1 ˇ2, r q ˇ 1 2 1 2 2 1 1;2 D 2 ˙ 2 and the following manipulations still hold formally, with the understanding that ˇ stands for ˇ1 and ˇ stands for ˇ2. Furthermore, we observe that

ˇˇ D ˇ1ˇ2 D 1:

21.3 Wiener-Hopf Factorization

The kernel K.s/ is an even function and it possesses 6 roots

K.s/ D 0 for s D˙s1; ˙s1 ; and s D˙{r1; (21.23) all of which are of multiplicity 1, in the general case. Here, s1 is taken to sit in the ﬁrst quadrant of the complex plane, v u s uÂ Ã Â Ã t 2 4 1 1; s1 D e C (21.24) R R p having let R D 2 e, with q 4 p 2 e D .1 /.3 1 C 2 2 2 C 1/ (21.25) 244 A. Nobili et al.

p p 1=4 and e 2 Œ 2. 5 2/ ;1is a well-known bending edge-wave constant [Ko60, KaEtAl14, KaNo15]. The double roots in the kernel K.s/ bring in four non-straight branch cuts which extend from s D˙1; ˙1 to ˙{1. It is observed that for D 0,wehave e D 1 and the roots ˙s1; ˙s1 coincide with the branch points for the double roots ˙ˇ; ˙ˇ, whence their multiplicity goes down to 1=2. Besides, it is easy to see that s1 is complex-valued for <R (subsonic 1 regime) and it sits in the complex plane on the circle of radius e , i.e. 1 .= /4 1 .{=2/; R : s1 D e exp tan D 2 .=R/ p In a similar fashion, letting S D 2 m, we ﬁnd the location of the purely imaginary roots ˙{r1 v u s uÂ Ã Â Ã t 2 4 1 1; r1 D m C C S S where we have deﬁned the monotonic decreasing function of q 4 p 2 m D .1 /.3 C 1 C 2 2 2 C 1/;

1 p 1=4 p p 1=4 Œ p .2 2 1/ ; 2. 5 2/ 1 We note that m 2 2 C and r is a real-valued 1 monotonic decreasing (increasing) function of (of ), whose minimum r1 D m is attained in the static case D 0, i.e. unlike s1, this root never reaches the real axis. In the special case of D 0 (stationary crack), we have p 2 1 .1 {/; r1 D e and s1 D C 2 e whence s1 sits on the bisector of the ﬁrst-third quadrants of the complex plane. In contrast, for >R (hypersonic regime) s1 turns real-valued and the root landscape (21.23) switches to

˙ s; ˙sC; ˙{r1; (21.26) where now 0

Let us define, for any value b >˛1 D max .Im.s1/; r1/, p p { { . / s b s C b . /; 0; F s D K s ¤ (21.28) c.s s1/.s C s1/.s s1 /.s C s1 / 21 Steady-State Crack Propagation 245 along with the constant c D 0.3 C /=4. In the special case D 0, we need set p p s {b s C {b F.s/ D p p K.s/; D 0: p p c s s1 s C s1 s s1 s C s1 .s {r1/.s C {r1/ The function F.s/ is deprived of zeros in a semi-infinite strip of analyticity about the real axis S, extending along the imaginary axis up to ˛1 D ˛2 D Im.s1/, and it is such that limjsj!1 F.s/ D 1 in this strip. The Cauchy integral theorem gives I 1 ln F.z/ ln F.s/ D dz; 2 { C z s where C may be taken as the close path in the analyticity strip consisting of two parallel infinite lines a little above and a little below the real axis while s sits within this closed path. The former contribution brings along a minus function, F.s/,the latter a plus function, FC.s/, for we may define

~~FC.s/ D exp R.s/ and F.s/ D FC.s/; where Z 1 1{c ln F.z/ R.s/ D dz;0~~

~~F.s/ D FC.s/F.s/; and the system (21.22) reads~~

vC 1 N0 K .s/wN 0 C D {V0 ; KC.s/ .s C a/KC.s/ (21.30a) p p p p . / ˇ ˇ ˇ ˇ K s s C s C s C s p p N0 mN 0 D{M0 ; s ˇ s C ˇ KC.s/ .s C a/KC.s/ (21.30b) where p .s ˙ s1/.s s / K˙.s/ D c p 1 F˙.s/ exp.˙{ =4/; s ˙ {b with the property that K.s/ D KC.s/. Clearly, for large values of jsj, we get the asymptotic behavior p K˙.s/ c exp.˙{ =4/jsj3=2: 246 A. Nobili et al.

Finally, the RHS are split in terms of a plus and a minus function, thus giving Â Ã vC 1 1 N0 V0 V0 { D { K .s/wN 0 ; KC.s/ s C a KC.s/ K.a/ .s C a/K.a/ (21.31a) p p p p p p ! s C ˇ s ˇ M0 s C ˇ s ˇ a ˇ a C ˇ mN C { KC.s/ 0 s C a KC.s/ K.a/ p p ˇ ˇ . / M0 a a C K s D { C p p N0 ; (21.31b) s C a K.a/ s ˇ s C ˇ

Since the LHSs (RHSs) represent two analytic functions in the upper (lower) half complex plane with a common strip of regularity, they can be analytically continued to the whole complex plane giving two entire functions E1.s/ and E2.s/, i.e. they are holomorphic over the whole complex plane. It is observed that both hands of 1 Equation (21.31) behave like s as s !1, whereupon E1.s/ Á E2.s/ Á 0,by Liouville’s theorem. Indeed,

w.x/ x3=2 )Nw.s/ s5=2; .x/ x1=2 ) N .s/ s3=2; m.x/ x1=2 )NmC.s/ s1=2;v.x/ x3=2 )NvC.s/ s1=2; the latter being meaningful in a distributional sense. Thus 1 V0 wN 0 D { ; (21.32) K.a/ .s C a/K.s/ and p p p p ˇ ˇ ˇ ˇ a a C s s C N0 D {M0 : (21.33) K.a/ .s C a/K.s/

Likewise, we obtain a direct expression for the unilateral Fourier transform of bending moment and shearing force along the co-ordinate axis y D 0, namely p p ! ˇ ˇ C. / C M0 a a C K s mN 0 D { 1 p p (21.34) s C a s C ˇ s ˇ K.a/ and Â Ã C V0 K .s/ vNC D { 1 : (21.35) 0 s C a K.a/ 21 Steady-State Crack Propagation 247

~~It is observed that, according to Jordan’s lemma [Ro69], Equations (21.32) and (21.33) satisfy both BCs (21.9) and, by the same argument, Equations (21.35) and (21.34) convey the conditions (21.10).~~

~~References~~

[AnEtAl63] Ang, D.D., Folias, E.S., Williams, M.L.: The bending stress in a cracked plate on an elastic foundation. J. Appl. Mech. 30(2), 245–251 (1963) [KaNo15] Kaplunov, J., Nobili, A.: The edge waves on a Kirchhoff plate bilater- ally supported by a two-parameter elastic foundation. J. Vib. Control (2015). doi:10.1177/1077546315606838 [KaEtAl14] Kaplunov, J., Prikazchikov, D.A., Rogerson, G.A., Lashab, M.I.: The edge wave on an elastically supported Kirchhoff plate. J. Acoust. Soc. Am. 136(4), 1487–1490 (2014) [Ko60] Konenkov, Y.K.: A Rayleigh-type ﬂexural wave. Sov. Phys. Acoust 6, 122–123 (1960) [MiEtAl12] Mishuris, G., Piccolroaz, A., Radi, E.: Steady-state propagation of a mode III crack in couple stress elastic materials. Int. J. Eng. Sci. 61(0), 112–128 (2012) [No14] Nobili, A., Radi, E., Lanzoni, L.: A cracked inﬁnite Kirchhoff plate supported by a two-parameter elastic foundation. J. Eur. Ceram. Soc. 34(11), 2737–2744 (2014). Modelling and Simulation meet Innovation in Ceramics Technology [NoEtAl15] Nobili, A., Radi, E., Lanzoni, L.: On the effect of the backup plate stiffness on the brittle failure of a ceramic armor. Acta Mech. 227, 1–14 (2015) [PiEtAl12] Piccolroaz, A., Mishuris, G., Radi, E.: Mode iii interfacial crack in the presence of couple-stress elastic materials. Eng. Fract. Mech. 80, 60–71 (2012) [Ra08] Radi, E.: On the effects of characteristic lengths in bending and torsion on mode III crack in couple stress elasticity. Int. J. Solids Struct. 45(10), 3033–3058 (2008) [Ro69] Roos, B.W.: Analytic Functions and Distributions in Physics and Engineering. Wiley, New York (1969) Chapter 22 Mono-Energetic Neutron Space-Kinetics in Full Cylinder Symmetry: Simulating Power Decrease

~~F.R. Oliveira, B.E.J. Bodmann, M.T. Vilhena, and F. Carvalho da Silva~~

~~22.1 Introduction~~

Neutron space-kinetics in 3˚1 space-time dimensions has been addressed in several works (see, for instance, [DaEtAl01, GuEtAl05, GrHe07, MiEtAl02, AbNa06, AbNa07, AbHa08, AbHa09, Qu10]), where the more common approaches make use of numerical schemes in order to obtain a solution. Usually, Cartesian geometry is the preferred system, where also analytical schemes may be found as, for instance, in [CeEtAl15, PeEtAl14]. The same type of problem but cast in cylinder coordinates may be found in reference [Ha12]. Numerical solutions in one spatial dimension are presented in [WaEtAl10, Se14]. In this work we present an exact solution for the three dimensional spatial neutron diffusion kinetics problem. As a continuation in this line but for heterogeneous problems, a sectional homogeneous problem along the cylinder axis is considered. More speciﬁcally, we consider a domain composed of two cylinder sections with different nuclear parameter. The solution is determined for a mono-energetic model with one neutron precursor group using the technique of variable separation. Recalling that the multi-group extension of this problem is not self-adjoint, where variable separation does not apply, the present results may be considered a ﬁrst step in this direction, where a multi-group solution may be constructed from the mono-energetic solution and using a method like the Adomian

F.R. Oliveira () • B.E.J. Bodmann • M.T. Vilhena Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil e-mail: [email protected]; [email protected]; [email protected] F. Carvalho da Silva Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil e-mail: [email protected]

© Springer International Publishing AG 2017 249 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_22 250 F.R. Oliveira et al. decomposition, for instance. To this end the complete spectra with respect to each variable separation are analysed and truncated such as to provide an acceptable solution for this heterogeneous problem.

~~22.2 The Model~~

The system of partial differential equations that model neutron space-kinetics in cylindrical geometry is deﬁned by the contributions of mono-energetic neutrons and one group of delayed neutron precursors. The physically relevant domain is composed by two sectionally homogeneous cylinders with different nuclear parameters. Further, the space-kinetics model contains the usual terms of diffusion, removal, ﬁssion and neutron precursor decay.

~~1 @ rE; t D D rE; t ˙ rE; t C .1 ˇ/ ˙ rE; t C C rE; t v @t dif r f @ C rE; t DC rE; t C ˇ ˙ rE; t @t f~~

Here, is the Laplacian operator in cylinder coordinates, v is the neutron speed, ˙r is the macroscopic removal cross section, ˇ is the fraction of delayed neutrons, is the neutron precursor decay constant and ˙f is the macroscopic ﬁssion cross section times the neutron multiplication factor. The system is subject to the following initial and boundary conditions with rE D .r;;z/ 2 ˝cil D .0; R Œ ; Œ0; Z.

~~ ˇ ˙ ;0 ;0 f rE D 0 rE C rE D 0 rE (22.1) @ C . ;; ; / 0 . ;; ; / 0 . ;; ; / 0 limr!0 @r r z t D R z t D r Z t D .r;;0;t/ D 0 .r;;z; t/ D .r; C 2 ; 0; t/ (22.2)~~

Note that the equation together with initial and boundary conditions is invariant under scale transformation. Further, the homogeneous solution obeys the same boundary conditions as the heterogeneous problem, so that the base functions have formally the same structure in either homogeneous sub-domain, but the associated spectra are different due to different nuclear parameters. Continuity of the scalar ﬂux at the interface of the two sub-domains is obtained by matching the scales and a Fermat type reasoning implies conservation of the current density at the interface. 22 Mono-Energetic Neutron Space-Kinetics . . . 251

22.3 Variable Separation For convenience we deﬁne rE; t D F1 rE G1 .t/ and C rE; t D F2 rE G2 .t/, A D vDdif , B D v .1 ˇ/ ˙f ˙r, D D v, E D ˇ ˙f , H D and from E the second initial condition P DH . Separating variables and introducing the separation constants 1 and 2 we obtain relations and the equations to be solved.

~~ B 2 E F1 rE C F1 rE D 0 F2 rE D F1 rE A 1 d DE d G1 .t/ D 2G1 .t/ C G2 .t/ G2 .t/ D 1G1 .t/ C G2 .t/ (22.3) dt 1 dt~~

The spatial variables may be separated accordingly F1.r;;z/ D f .r/g./h.z/, where additional separation constants 3 and 4 are introduced. The equations to be solved in closed form (22.3) and (22.4) and subject to the initial (22.1) and boundary conditions (22.2)are

~~d2 1 d . / . / 3 . / 0; dr2 f r C r dr f r C f r D 2 d B2 . / 3 . / 0; (22.4) dz2 h z C A h z D d2 ./ 4 ./ 0: d2 g C g D~~

~~22.4 A Closed Form Solution in Cylinder Geometry~~

A detailed analysis of physically meaningful spectra for the separation constants results in solutions X X .rE; t/ D .n;k;l/.rE; t/ and C.rE; t/ D C.n;k;l/.rE; t/; (22.5) n;k;l n;k;l respectively, where n; k e l result from spectral analysis and associated eigenfunctions obtained for the spatial variables. One obtains as solution terms .n;k;l/ and C.n;k;l/ (for isotropic l D 0 and for anisotropic l >0separation conditions) ˛ Á .n;k;l/ n;l rE; t D .d1; ; ; exp .1; ; ; t/ C d2; ; ; exp .2; ; ; t// J r n k l n k l n k l n k l l R Â Ã k z .c1; sin .l/ C c2; cos .l// sin ; l l Z 252 F.R. Oliveira et al. Â Ã . / . / ˛ Á .n;k;l/ d1;n;k;l exp 1;n;k;lt d2;n;k;l exp 2;n;k;lt n;l C rE; t D E C Jl r 1;n;k;l H 2;n;k;l H R Â Ã k z .c1; sin .l/ C c2; cos .l// sin : l l Z

Here, Jl are the Bessel functions of the ﬁrst kind of order l. Further, note that i;n;k are determined from the spectrum of the space time separation and the ci;n;k;l, di;n;k;l and n;k;l are determined from the initial and boundary conditions. Â q Ã 1 2 1; ; ; H 2; ; ; .H 2; ; ; / 4 .2; ; ; H DE/ n k l D 2 C n k l C C n k l n k l Â q Ã 1 2 2; ; ; H 2; ; ; .H 2; ; ; / 4 .2; ; ; H DE/ n k l D 2 C n k l C n k l n k l " Â Ã # Á2 2 ˛n;l k 2; ; ; D B A C n k l R Z

Here, n 2f1;2;:::g, k 2f1;2;:::g and l 2f0; 1; 2; : : :g. The values ˛n;l >0are such that Jl .˛n;l/ D 0. For these solutions the following initial condition shall hold, ˛ Á .n;k;l/ .n;k;l/ n;l rE;0 D 0 rE D .d1;n;k;l C d2;n;k;l/ Jl r Â R Ã k z .c1; sin .l/ C c2; cos .l// sin ; l l Z where d1;n;k;l and d2;n;k;l are constants to be determined as shown below. C.n;k;l/ rE;0 D P .n;k;l/ rE;0 (22.6)

~~Further, equation (22.6) establishes a relation between the di;n;k;l (i 2f1; 2g). Ä Ä E E d2;n;k;l P D d1;n;k;l P ) d2;n;k;l D n;k;ld1;n;k;l 2;n;k;l H 1;n;k;l H and~~

~~P .2;n;k;l H/.1;n;k;l H/ E .2;n;k;l H/ n;k;l D E .1;n;k;l H/ P .2;n;k;l H/.1;n;k;l H/~~

Using the principle of superposition (22.5) of all components yields a closed form solution of the problem. The constants d1;n;k;0 may be determined upon applying the integral operator Z Z Z 2 Z R Á Á ˛ ;0 s z IŒ N1 Œ m D m;s;0 rJ0 r sin dr dz d 0 0 0 R Z 2 N ; ;0 .1 ; ;0/.RJ1 .˛ ;0// Z m; s m s D 2 C m s m 8 2 22 Mono-Energetic Neutron Space-Kinetics . . . 253 on the expression representing the initial condition.

~~1 ÄZ Z Á X 2 Z s z I N1 .1 / 0 rE D m;s;0 d1;n;k;0 C n;k;0 sin 0 0 Z n;kD1 Â Ã Z R Á Á k z ˛m;0 ˛n;0 sin dz d rJ0 r J0 r dr Z 0 R R~~

~~Z 2 c1;n;k;l sin .l/ C c2;n;k;l cos .l/ d D 0 0~~

This implies ( X1 . .˛ //2 1 RJ1 m;0 I 0 r N d1; ; ;0 .1 ; ;0/ E D m;s;0 m k C m k 2 kD1 Z Z Á Â Ã 2 Z s z k z sin sin dz d 0 0 Z Z by the orthogonality property of Bessel functions of the ﬁrst kind of order 0 with respect to the identity weight function. One obtains then Z Z Z Â Ã 2 Z R Á ˛ ;0 k z N1 Œ n : d1;n;k;0 D n;k;0 0 rE rJ0 r sin dr dz d 0 0 0 R Z

Finally, one determines c1;n;k;l and c2;n;k;l, following an analogue procedure. Z Z Z Â Ã 2 Z R Á ˛ ; k z P1 n l . / ; c1;n;k;l D n;k;l 0 rE rJl r sin l sin dr dz d 0 0 0 R Z Z Z Z Â Ã 2 Z R Á ˛ ; k z P1 n l . / ; c2;n;k;l D n;k;l 0 rE rJl r cos l sin dr dz d 0 0 0 R Z

P .1 /. .˛ //2 where n;k;l D 4 C n;k;l RJlC1 n;l Z. So far the structure of the solution was determined which is valid for either of the cylinder segments, i.e. the upper z 2 ..1 /Z; Z or the lower z 2 Œ0; .1 /Z/ with 0<<1. As mentioned earlier both solutions are invariant under scale transformation, so that the arbitrary scale may be used to match the solutions at the interface z D .1 /Z.Letthe solutions in the respective cylinder elements be Œ1.rE; t/; if 0 Ä z Ä .1 /Z .rE; t/ D ; Œ2.rE; t/; if .1 /Z Ä z Ä Z 254 F.R. Oliveira et al. then the scale degree of freedom allows to satisfy

~~Œ1 .r;;.1 /Z; t/ D Œ2 .r;;.1 /Z; t/ ;~~

Œ1 @ Œ2 @ D Œ1.rE; t/ DD Œ2.rE; t/; dif @z dif @z where numerical results are shown in the next section. Note that the second scale degree of freedom present in the problem may be compared to solving problems with adjacent media of different physical properties by Fermat’s principle.

~~22.5 Numerical Results~~

As a case study we consider a simulation of the effect from inserting a control rod and calculate the solution for positions of the rod end for 2f0:2; 0:4; 0:6; 0:8g. Here indicates the position of the inserted end as a fraction of the cylinder height. The nuclear parameters for the numerical simulations are given in Table 22.1, where region 2 corresponds to the cylinder segment where larger absorption due to the presence of the control rod occurs. For our obtained solution we present some numerical results. In Figure 22.1 we show the scalar neutron flux for four moments of control rod insertion and for an arbitrary angle (here D 4 ), the first one where the rod is at a position corresponding to 20% of the cylinder height, the second with 40%, the third with 60% and the last one where the rod occupies 80% of full insertion. In Figure 22.2 the time dependence of the scalar flux integrated over the radial and angular variables is shown. In Figure 22.3, we show the time dependence of the scalar flux integrated all over domain. The conserved current density is numerically verified in Table 22.2 and as expected continuity for the scalar flux as well as the current density holds.

~~lim!0 Œ .t; r;;0:5Z C / D .t; r;;0:5Z / E . ; / Œ1 @ Œ1. ; / E . ; / Œ2 @ Œ2. ; / j1 rE t DDdif @z rE t and j2 rE t DDdif @z rE t~~

~~Table 22.1 Nuclear parameters for the 2 regions~~

~~Ddif v ˙a ˙f ˇ Reg. 1 0.96343 1:1035 107 0.015843 0.033303 0.08 0.0065 Reg. 2 0.96343 1:1035 107 0.115843 0.033303 0.08 0.0065 22 Mono-Energetic Neutron Space-Kinetics . . . 255~~

~~-5 x10~~

~~1.2~~

~~) t~~

~~, , 0.8~~

~~z p~~

~~4 0.6~~

~~, , , , r~~

~~( 0.4 f 0.2 0 10 8 9 10 6 7 8 4 6 4 5 r [cm] 2 2 3 z [cm] 0 0 1~~

~~-5 x10~~

~~1.2~~

)

~~t , ,~~

~~z 0.8~~

~~p 4 , , , ,~~

~~r 0.6~~

~~( f 0.4~~

~~0.2~~

~~0 10 8 6 10 8 9 4 6 7 2 4 5 r [cm] 2 3 0 0 1 z [cm]~~

~~-5 x10~~

~~1.2~~

~~) t~~

~~, , 0.8~~

~~z p~~

~~4 0.6~~

~~, , , ,~~

~~r (~~

~~f 0.4~~

~~0.2~~

~~0 10 8 6 9 10 7 8 4 5 6 2 4 r [cm] 2 3 z [cm] 0 0 1~~

~~-5 x10~~

~~1.2~~

~~0.8~~

)

~~t , ,~~

~~z 0.6~~

~~p 4 , , , ,~~

~~r 0.4~~

~~( f 0.2~~

~~0 10 8 6 4 10 2 7 8 9 4 5 6 r [cm] 0 1 2 3 0 z [cm]~~

~~Fig. 22.1 Effect of the insertion of the control rod on the scalar neutron ﬂux for positions with D 0:2, D 0:4, D 0:6 and D 0:8 256 F.R. Oliveira et al.~~

~~×10-3 3~~

~~20% 2.5 40% 80% 2 ) z~~

~~( 1.5 p~~

~~0.5~~

~~0 0 12345678910 Z [ cm ]~~

~~Fig. 22.2 Linear power density along the cylinder axis for instances corresponding to three positions with D 0:2, D 0:4 and D 0:8~~

~~0.0104783~~

~~0.0104782~~

~~0.0104781 80%~~

~~0.010478~~

~~0.0104779 ) t (~~

~~ρ 0.0104778~~

~~0.0104777~~

~~0.0104776~~

~~0.0104775~~

~~0.0104774 0 5 10 15 20 25 t[ s]~~

~~Fig. 22.3 Time evolution of the linear power density~~

Table 22.2 Numerical Ej1jzD0:5Z Ej2jzD0:5Z values for the current t D 1s 8.03053522862e-07 8.03053522865e-07 densities Ej1 and Ej2,for different times and with t D 5s 5.83799858347e-07 5.83799858389e-07 Z R 10 z D 2 , r D 2 and D 4 t D s 3.91889240654e-07 3.91889240795e-07 t D 15s 2.63064764268e-07 2.63064764463e-07 22 Mono-Energetic Neutron Space-Kinetics . . . 257

~~22.6 Conclusion~~

In the present work we determined an analytical solution for the mono-energetic neutron space-kinetics equation with one precursor concentration in cylindrical geometry. The solution was constructed by the use of variable separation, which was applied to a scenario with two adjacent homogenized cylindrical cells with different nuclear parameters in each section. The choice of the parameter sets is to mimic the insertion of a control rod. As a result we present the scalar angular flux with its space-time dependencies. Considering the heterogeneity of the domain, the encountered results are in agreement with what is expected for such a scenario. One of the peculiarities of the present problem is that initial as well as boundary conditions maintain scale invariance of the solution of the considered space-kinetics problem. This scale degree of freedom allows to construct the solution of each homogeneous cylinder cell using the same functional structure where the difference in the sectional solution comes from the fact that the spectrum for each cell is different due to the differences in its physical properties, while continuity of the scalar flux is obtained by matching the flux magnitudes at the interface. As a matter of fact, the current density depends besides the flux magnitude also on a typical scale related to the eigenvalues, i.e. the spectrum in either cylinder segment, so that fixing this scale implies conservation of the current density across the interface. It is noteworthy that the coupling at the interface does not appear explicitly in the established formalism, but is attained by stretching or contracting the afore mentioned scales where the second scale resembles Fermat’s principle. For future work, we focus on an extension to a multi-group description, starting from the mono-energetic solution and correcting this solution by the out- and in- scattering terms, that establishes the links between the energy groups and may be implemented by the use of a decomposition method. These considerations are topics of future works, where we also increase the number of sections such as fuel region, moderator and reflector regions besides the region influenced by the presence of the control rod.

~~References~~

[AbHa08] Aboanber, A.E., Hamada, Y.M.: Generalized Runge-Kutta method for two- and three- dimensional space-time diffusion equations with a variable time step. Ann. Nucl. Energy 35(6), 1024–1040 (2008) [AbHa09] Aboanber, A.E., Hamada, Y.M.: Computation accuracy and efﬁciency of a power series analytic method for two- and three-space-dependent transient problems. Prog. Nucl. Energy 51(3), 451–464 (2009) [AbNa06] Aboanber, A.E., Nahla, A.A.: Solution of two-dimensional space-time multigroup reactor kinetics equations by generalized Padé and cut-product approximations. Ann. Nucl. Energy 33(3), 209–222 (2006) 258 F.R. Oliveira et al.

[AbNa07] Aboanber, A.E., Nahla, A.A.: Adaptive matrix formation AMF method of space-time multigroup reactor kinetics equations in multidimensional model. Ann. Nucl. Energy 34(1–2), 103–119 (2007) [CeEtAl15] Ceolin, C., Schramm, M., Vilhena, M.T., Bodmann, B.E.J.: On the neutron multi- group kinetic diffusion equation in a heterogeneous slab: an exact solution on a ﬁnite set of discrete points. Ann. Nucl. Energy 76, 271–282 (2015) [DaEtAl01] Dahmani, M., Baudron, A.M., Lautard, J.J., Erradi, L. : A 3D nodal mixed dual method for nuclear reactor kinetics with improved quasistatic model and a semi- implicit scheme to solve the precursor equations. Ann. Nucl. Energy 28(8), 805–824 (2001) [GrHe07] Grossman L.M., Hennart, J.P.: Nodal diffusion methods for space-time neutron kinetics. Prog. Nucl. Energy 49(3), 181–216 (2007) [GuEtAl05] Gupta, A., Modak, R.S., Gupta, H.P., Kumar, V., Bhatt, K.: Parallelised Krylov subspace method for reactor kinetics by IQS approach. Ann. Nucl. Energy 32(15), 1693–1703 (2005) [Ha12] Hançerliogullari,ˇ A.: Neutronic calculations at uranium powered cylindrical reactor by using Bessel differential equation. In: Korkut, T. (ed.) Nuclear Science and Technology, pp. 15–24. Transworld Research Network, Kerala (2012) [MiEtAl02] Miró, R., Ginestar, D., Verdú, G., Hennig, D.: A nodal modal method for the neutron diffusion equation. Application to BWR instabilities analysis. Ann. Nucl. Energy 29(10), 1171–1194 (2002) [PeEtAl14] Petersen, C.Z., Bodmann, B.E.J., Vilhena, M.T., Barros, R.C.: Recursive solutions for multi-group neutron kinetics diffusion equations in homogeneous three-dimensional rectangular domains with time dependent perturbations. Kerntechnik 79, 494–499 (2014) [Qu10] Quintero-Leyva, B.: The multi-group integro-differential equations of the neutron diffusion kinetics. Solutions with the progressive polynomial approximation in multi- slab geometry. Ann. Nucl. Energy 37(5), 766–770 (2010) [Se14] Seliverstov, V.V.: Kinetic diffusion equation in a one-dimensional cylindrical geometry. Atom. Energy 115, 319–327 (2014) [WaEtAl10] Wang, D., Li, F., Guo, J., Wei, J., Zhang, J., Hao, C.: Improved nodal expansion method for solving neutron diffusion equation in cylindrical geometry. Nucl. Eng. Des. 240, 1997–2004 (2010) Chapter 23 Asymptotic Solutions of Maxwell’s Equations in a Layered Periodic Structure

~~M.V. Perel and M.S. Sidorenko~~

~~23.1 Introduction~~

The Maxwell equations with rapidly oscillating coefficients have applications, for example, in the description of propagation of electromagnetic waves in photonic crystals. Homogenization (or averaging) methods make it possible to reduce the problem to the solution of a problem for equations with constant coefficients, which are found by some averaging of the initial coefficients. The principal order of the asymptotics is constructed under the assumption that kb ! 0, where k is the wave number and b is the period of the medium. The neighborhood of the minimum of the dispersion function contributes to the averaged problem. In what follows, we deal with a different problem, which resembles the averaging methods. Here we present the main results of [PeSi15], partially announced in [PeSi16]. Some of their applications and further discussions can be found in [SiPe12] and [PeSi11]. We assume that the frequency is in the neighborhood of some stationary point of the dispersion function; stationary points exist on all the leafs of this function. Instead of the condition kb ! 0, we use the condition kb 1=2 ."avav/ , where "av and av characterize the orders of " and , respectively. The small parameter in our case is the ratio of the period of the medium to the parameter characterizing the variation of the solutions in planes orthogonal to the direction of variation of " and . We solve the problem by the method of two-

M.V. Perel () St. Petersburg State University, St. Petersburg, Russia Ioffe institute, St. Petersburg, Russia e-mail: [email protected] M.S. Sidorenko St. Petersburg State University, St. Petersburg, Russia e-mail: [email protected]

© Springer International Publishing AG 2017 259 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_23 260 M.V. Perel and M.S. Sidorenko scale asymptotic expansion. The principal order of the asymptotics is found to be a linear combination of two linearly independent periodic solutions with slowly varying coefﬁcients that satisfy a system of differential equations with constant coefﬁcients, which is an analogue of the averaged system. We solve this system by the Fourier transformation method and use the results to solve a boundary value problem for the Maxwell equations in a half-space. The boundary data vary slowly, and the slowness of their variation determines the small parameter of the solution expansion. We supplement the boundary data with the limiting absorption principle to ensure a unique solution.

~~23.2 Maxwell’s Equations and the Floquet–Bloch Solutions~~

Consider a stratified periodic medium with dielectric permittivity ".z/ and magnetic permeability .z/; which are piecewise smooth. A monochromatic electromagnetic field E, H in such a medium satisfies the Maxwell equations

curl E D ikH; (23.1) curl H Dik"E; where k D !=c, ! is the frequency, c is the speed of light, ".z C b/ D ".z/, .z C b/ D .z/, ".z/ ¤ 0; and .z/ ¤ 0. The Floquet–Bloch solutions Â Ã Â Ã E . / E D ei pxxCpyy eipzz (23.2) H H are well-known solutions of (23.1). We call px; py the lateral components of the wave vector, and pz the quasi-momentum determined from the condition that E.z C b/ D E.z/, H.z C b/ D H.z/ for a given frequency !; that is, pz D pz.px; py;!/. The values of ! for which pz is complex belong to forbidden zones (or gaps); the values of ! corresponding to real pz belong to allowed zones. The inverse function ! D !.p/ is a multivalued function of the real variables p D .px; py; pz/, where px 2 .1; 1/, py 2 .1; 1/, and pz 2 . =b; =b/. We call it the dispersion function. Therefore, E and H, which depend on px, py, and !, can be expressed as functions of p. We note that there are two types of solutions for every p, called the transverse electric (TE) and transverse magnetic (TM) types. The corresponding dispersion functions are denoted with superscripts E or H; that is, the dispersion relation for the TE and TM solutions is written as ! D !E.p/ and ! D !H.p/, respectively. When px D py D 0, the concepts of TE and TM polarizations lose their meaning. Each solution of the system of two ordinary differential equations

~~dE0 dH0 i DkH0; i Dk"E0 (23.3) dz dz 23 Asymptotic Solutions of Maxwell’s Equations 261 enables us to determine two independent solutions of the Maxwell equations, namely~~

~~X t X t E D .E0;0;0/; H D .0; H0;0/; (23.4) Y t Y t E D .0; E0;0/; H D .H0;0;0/:~~

~~23.3 Two-Scale Solutions of the Maxwell Equations~~

~~In this section we consider the solutions of the Maxwell equations which depend on distance variables of two scales: x; y; z, having a scale of period b, and slow variables~~

~~ Á x;Á y;Á z; D .;;/; where 1: We seek particular solutions of the form~~

~~E.z; / D E.z; /eipzz; H.z; / D H.z; /eipzz; (23.5) X E.z; / D nE.n/.z; /; E.n/.z C b; / D E.n/.z; /; n0 X (23.6) H.z; / D nH .n/.z; /; H .n/.z C b; / D H .n/.z; /: n0~~

~~We make two additional assumptions. First, we assume that the frequency of the electromagnetic ﬁeld is close to the frequency of a stationary point of some leaf of the dispersion function !:~~

~~2 ! D ! C ı!; ı! 1; where ˇ r!f ˇ D 0; ! D !f .p /: p ~~

Here, the superscript f stands for either E or H. It is shown that at the stationary points, the dispersion functions of both polarizations touch each other; that is, H E ! .p/ D ! .p/ and px D py D 0, pz D˙ =b;0: At the stationary points, the Floquet–Bloch solutions (23.2) do not depend on x and y, they satisfy a system of ordinary differential equations, and the frequency ! corresponds to the edge of the forbidden zone. We note that for every px, py, and !, there are two linearly independent Floquet–Bloch solutions corresponding to pz D˙pz.px; py;!/. At the point px D 0, py D 0, ! D !, these solutions may be linearly dependent, in which case the second independent Floquet–Bloch solution is unbounded. This unboundedness is our second assumption. 262 M.V. Perel and M.S. Sidorenko

We insert (23.5) and (23.6) into Maxwell’s equations and obtain a system that is solved recurrently; details can be found in [PeSi15]. Finally, we obtain the principal order of the asymptotics of the solutions and establish that the successive approximation equations are solvable. The asymptotics of the solutions are expressed through the bounded Floquet– Bloch solution (23.4)of(23.3), which at the stationary point is marked by an asterisk f f subscript E and H; where f D X or f D Y. The principal term is Â Ã Â Ã Â Ã X Y E . ; / ˛ ./ E . ; / ˛ ./ E . ; /; z ' 1 X z C 2 Y z (23.7) H H H where ˛j./; j D 1; 2; are solutions of the system of differential equations

@2˛ @2˛ @2˛ @2˛ 1 H 1 E 1 2 H E !R C !R C !R C 2.ı!/˛1 .!R R! / D 0; @2 k @2 k @2 ? @@ k k @2˛ @2˛ @2˛ @2˛ 2 E 2 H 2 1 H E !R C !R C !R C 2.ı!/˛2 .!R R! / D 0: @2 k @2 k @2 ? @@ k k (23.8) The coefﬁcients of these equations are the second-order derivatives of the dispersion functions at the stationary point:

~~@2!f @2!f @2!f !R f D j D j ; !R D j ; f D H; E: k @ 2 p @ 2 p ? @ 2 p px py pz~~

We note that the derivatives in the direction of the layers depend on the type of the wave. The derivatives in the direction normal to the layers for both dispersion functions coincide, so the notation !R ? does not need a superscript.

~~23.4 Boundary Value Problem in a Half-Space~~

We now consider a boundary value problem for the Maxwell equations in a half- space (or half-plane) in the case where the stationary point is a saddle point. Speciﬁcally, we impose the boundary conditions

~~H1jzD0 D g2.; /; H2jzD0 D g1.; /: (23.9)~~

We require that the functions gj, j D 1; 2; be infinitely differentiable, and that these functions and their derivatives of any order decrease at infinity faster than any power of 2 C 2 (in other words, they belong to the Schwartz class). Then the Fourier transforms gOj of gj also belong to the Schwartz class. Additionally, we require that 2 2 the functions gOj.p , p/=.p C p/ be infinitely differentiable. 23 Asymptotic Solutions of Maxwell’s Equations 263

~~We impose the limiting absorption principle; that is, we assume that~~

~~Im ı! > 0; and require that~~

2 max jHj!0; 2 max jEj!0; (23.10) !1 !1 where ÁjjD2 C 2 C 2.Weshowedin[PeSi15] that conditions (23.9) and (23.10) ensure the uniqueness of the solution of the Maxwell equations. We also assume that the bounded Floquet–Bloch solution of (23.3) satisﬁes the condition H0.0/jp ¤ 0: The solutions of equations (23.8), which generate the principal order of the asymptotics (23.7) satisfying conditions (23.9) and (23.10), were constructed in [PeSi15] by the Fourier transformation method. They are

~~Z1 Z2 1 ˛ ./ ipk. cos C sin / 1 D 2 dpkpk d e .2 / H0.0/jp 0 0 ip 2 ip 2 e H cos bg1 cos sin bg2 C e E sin bg1 C sin cos bg2 ; (23.11)~~

Z1 Z2 1 ˛ ./ ipk. cos C sin / 2 D 2 dpkpk d e .2 / H0.0/jp 0 0 ip 2 ip 2 e H . cos sin bg1 C sin bg2/ C e E . cos sin bg1 C cos bg2/ ; (23.12) where q q 2 2 2; 2 2 2; pH D Hpk C m pE D Epk C m !R H !R E ı! 2 k ;2 k ; 2 2 ; H D E D m D !R? !R ? !R? and we have taken the root branches with positive real parts. We remark that in the 2 ;2 >0 ı! > 0 ˛ ./; case of a saddle point we have H E .If , the coefﬁcients j j D 1; 2; are represented as a sum of two functions of slow variables, which are the . / . / integral superposition of eipk cos C sin eipH and eipk cos C sin eipE : Each of these functions satisﬁes the Klein–Gordon–Fock equation, or the wave equation if ı! D 0. The speeds of propagation of the waves described by these equations are H and E: This result has a physical implication, showing that the electromagnetic 264 M.V. Perel and M.S. Sidorenko waves of the TM and TE polarizations propagate in the periodic structure inside two cones with cone angle dependent on H and E; respectively, if the boundary data have compact support.

~~23.5 Conclusions~~

We have constructed an asymptotic expansion of the solution of the Maxwell equation in a medium periodic in the z-direction. The principal term of the asymptotics is a linear combination of two particular Floquet–Bloch solutions periodic in z. The coefﬁcients are smooth envelope functions and vary slowly compared to the period b. For our speciﬁc boundary value problem, the small parameter appears in the boundary data and is the ratio of the period of the medium to the parameter characterizing the boundary data variation. Equations (23.8)forthe envelope functions have been solved (see (23.11) and (23.12)) and their solution is represented as a sum of two terms satisfying the Klein–Gordon–Fock equations or wave equations. Each of the wave equations has its own propagation speed. For smooth boundary data with compact support, this fact determines two abrupt fronts of the solution inside the medium. This phenomenon is analogous to conical refraction.

~~Acknowledgements The work of M.V.P. was supported by SPbGU grant 11.42.1073.2016.~~

~~References~~

[PeSi11] Perel, M.V., Sidorenko, M.S.: Effects associated with a saddle point of the dispersion surface of a photonic crystal. In: Days on Diffraction (DD), 2011, pp. 145–148. IEEE, St. Petersburg (2011) [PeSi15] Perel, M.V., Sidorenko, M.S.: Two-Scale Approach to an Asymptotic Solution of Maxwell Equations in Layered Periodic Medium. Submitted. Preprint arXiv:1511.00115 [math-ph] (2015) [PeSi16] Perel, M.V., Sidorenko, M.S.: Asymptotic study of a two-scale electromagnetic ﬁeld in a layered periodic structure. In: Days on Diffraction (DD), 2016, pp. 319–322. IEEE, St. Petersburg (2016) [SiPe12] Sidorenko, M.S., Perel, M.V.: Analytic approach to the directed diffraction in a one- dimensional photonic crystal slab. Phys. Rev. B 86(3), 035119 (2012) Chapter 24 Some Properties of the Fractional Circle Zernike Polynomials

~~M.M. Rodrigues and N. Vieira~~

~~24.1 Introduction~~

The cornea is the major refracting component of the human eye, contributing approximately two thirds of the eye optical power. There are many types of mathematical models for corneal topography. The most common and simple models are based on conic sections [KaIs06], or on generalized conic functions [ScEtAl95]. Currently, the Zernike polynomials are the standard functions for the description of the wave front aberrations of the human eye and have been also used in the modeling of cornea surfaces [IsEtAl02]. The aim of this paper is to introduce the so-called fractional circle Zernike polynomials and study some of their properties. The outline of the paper reads as follows: In Section 24.2, we recall some basic notions about circle Zernike polynomials, fractional calculus, and g-Jacobi functions. Then, we introduce the fractional circle Zernike polynomials and the explicit formulas for these new polynomials will be present, as well as the study of the existence of a continuous orthogonality relation will be done. Moreover, we present some recurrence relations for consecutive and distant neighborhoods, and we study the differential properties of these fractional polynomials. In the last section, we present a graphical representation for some fractional circle Zernike polynomials.

M.M. Rodrigues ()•N.Vieira CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal e-mail: [email protected]; [email protected]

~~© Springer International Publishing AG 2017 265 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_24 266 M.M. Rodrigues and N. Vieira~~

~~24.2 Preliminaries~~

We start recalling some basic facts about circle Zernike polynomials which are studied in several references, see, for instance, [BoWo59]. The odd and even Zernike polynomials are given by (see [PrRu89])

m.; / o m.; / m./ . /; Zn D Un D Rn sin m m.; / e m.; / m./ . /; Zn D Un D Rn cos m where is the azimuthal angle with 0 Ä Ä 2 , is the radial distance with 0 1 m./ ; Ä Ä , and the radial function Rn is deﬁned for n m 2 0, with m Ä n by 8 ˆ nm <ˆ X2 .1/k .n k/Š n2k; for n m even; Rm./ D Š nCm Š nm Š n ˆ k 2 k 2 k :ˆ kD0 0; for n m odd:

~~m./ When n m is even, the radial functions Rn can be written in terms of the Jacobi .˛;ˇ/ polynomial Pn .x/ as~~

~~nm . ;0/ m 2 m m 2 R ./ D .1/ P nm .1 2 /: n 2~~

Now we recall the deﬁnitions of Riemann-Liouville fractional integral and fractional derivative in the sense of Riemann-Liouville (see [Po99, SaEtAl93]). Deﬁnition 1 For t >0 Z 1 t J f .t/ WD .t / 1 f ./ d . / 0 is called the Riemann-Liouville fractional integral of the function f .t/ of order 2 ¼ with Re. / > 0.

~~Deﬁnition 2 If t >0and m 2 0 such that m 1 Ä ~~

~~Df .t/ D Dm ŒJmf .t/ ; (24.1)~~

(if it exists) where m >0. Now, we pass to the deﬁnition and basic properties of the g-Jacobi functions introduced and studied in [MiEtAl07]. These functions will play a key role in the deﬁnition of the fractional Zernike polynomials, and it corresponds to a generalization for the fractional case of the classical Jacobi polynomials. 24 Fractional Circle Zernike Polynomials 267

Definition 3 We define the (generalized or) g-Jacobi functions by the formula h i .˛;ˇ/ 1 ˛ ˇ C˛ Cˇ P .t/ D .2/ . C 1/ .1 t/ .1 C t/ D .1 t/ .1 C t/ ; (24.2) where 2 C, ˛>1, ˇ>1 and D is the Riemann-Liouville fractional differential operator (24.1). In the following results we recall some properties of the g-Jacobi functions (see [MiEtAl07]), which are analogous to the corresponding properties of the classical Jacobi polynomials. In the first two results we give the corresponding explicit formulas for these functions. Theorem 1 (Explicit Formula) For the g-Jacobi functions holds the explicit formula Â ÃÂ Ã X1 .˛;ˇ/ C ˛ C ˇ P .t/ D 2 .t 1/k .t C 1/ k; (24.3) k k kD0 where Â Ã ˛ .1C ˛/ D ˇ .1C ˇ/ .1 C ˛ ˇ/ is the binomial coefficient with real arguments. Theorem 2 (Explicit Formula) The g-Jacobian functions can be represented as ! Â Ã .˛;ˇ/ C ˛ 1 t P .t/ D 2F1 ; C ˛ C ˇ C 1I ˛ C 1I 2 ! 1 Â Ã 1 X .1C C ˛ C ˇ C k/.1C ˛ C / t 1 k D ; (24.4) .1C / k .1C C ˛ C ˇ/ .1 C ˛ C k/ 2 kD0 where 2F1.a; bI cI t/ is the Gauss hypergeometric function. In the following results we will present the most basic properties of the g-Jacobi functions. Theorem 3 (Differential Equation) The g-Jacobi functions satisfy the linear homogeneous differential equation of second order

~~.1 t/2y00 C Œˇ ˛ .˛ C ˇ C 2/t y0 C . C ˛ C ˇ C 1/y D 0; or ˚ « ˛C1 ˇC1 0 ˛ ˇ Dt .1 t/ .1 C t/ y C . C ˛ C ˇ C 1/.1 t/ .1 C t/ y D 0: 268 M.M. Rodrigues and N. Vieira~~

Theorem 4 The g-Jacobi functions satisfy the following properties (with n 2 ): .˛;ˇ/ .˛;ˇ/ 1. lim !n P .t/ D Pn .t/; .˛;ˇ/ .˛;ˇ/ 2. P .t/ DÂ.1/ PÃ .t/; ˛ .˛;ˇ/.1/ C 3. P D ; Â Ã ˇ .˛;ˇ/. 1/ C 4. P D ; .˛;ˇ/. / 1 . ˛ ˇ 1/ .˛C1;ˇC1/. / 5. DtP t D 2 C C C P 1 t .

~~24.3 Fractional Circle Zernike Polynomials~~

The aim of this section is to introduce the so-called fractional circle Zernike polynomials and study some of their main properties. Taking into account the definition of g-Jacobi functions presented previously and the ideas presented in [BoWo59], we introduce the definition for the fractional circle Zernike polynomials. Definition 4 We define the fractional circle Zernike polynomials by

m m Z .; / D R ./ sin.m/; when Œ m is even ; m m Z .; / D R ./ cos.m/; when Œ m is odd ; where m 2 0, 2 C, >m, 0 Ä Ä 1 is the radial distance, 0 Ä Ä 2 is the azimuthal angle, and Œˇ represents the integer part of ˇ. The fractional radial m function R ./ will be deﬁned as

~~m . ;0/ m Œ 2 m m 2 R ./ D .1/ P m .1 2 /; (24.5) 2~~

.˛;ˇ/ where P .x/ are the g-Jacobi functions deﬁned in (24.2). Taking into account the properties of the g-Jacobi functions introduced in the paper [MiEtAl07], we derive the following results. m Theorem 5 (Explicit Formula) The fractional radial function R ./ allows the following explicit formulas Â ÃÂ Ã X1 Cm m Œ vm m 2 2 m Rm./ D 2 2 .1/ 2 Ck 2 2 kCm .1 / 2 k m C k k kD0 2 Â Ã X1 m 1 Cm 1 C k C 2 Œ m 2 D 2 .1/ 2 Ck mC k; 1 C m k .m C k/Š 2 kD0 24 Fractional Circle Zernike Polynomials 269 where Â Ã ˛ .1C ˛/ D : ˇ .1C ˇ/ .1 C ˛ ˇ/

Proof The result is obtained by straightforward calculations based on the substitution of (24.5) into the explicit formulas (24.3), (24.4), respectively. m Now, we present some recurrence relations for R ./. We will split our relations in two results. The ﬁrst one refers to the recurrence relations for the consecutive neighborhoods, and the second for distant neighborhoods. m Theorem 6 The radial function R ./ veriﬁes the following recurrence relations for consecutive neighborhoods: 2 m m1 m R ./ D R 1 ./ C R 2./I m 2. 2/ 2 m C m1 C m C m R ./ D R 1 ./ C R 2./I m C 2 m C 2 C 2 2. 2/ m m C m C m1 R ./ D R 2./ R 1 ./I C m C 2 C C m C 2 2. 3/ Œ 2 .1 22/. 2/. 4/ m C m C C C m R ./ D R 2./ . C m C 2/ . m C 2/ . C 4/ C . /. 4/. 2/ m C m C C m R 4./I . C m C 2/. m C 2/. C 4/ C 2. 1/ Œ 2 .1 22/. 2/ m m C m R ./ D R 2./ . 2 m2/. 2/ . 2/. 1/ C m m m R 4./I . 2 m2/. 2/ 2.1 / .2 / .1 2/ m C m C C m1 R ./ D R 1 ./I . C m/ 2 .1 2/ 2. 22 1/ . 4/ m m C m mC1 C m C mC2 R ./ D R ./ R ./: C m 2 C1 C m C 2 C2

Proof The proof of the recurrence relations takes into account the recurrence relations presented in [AbSt72]-Chapter 22 and [PrEtAl88]-Appendix II.11 for the classical Jacobi polynomials, the deﬁnition of g-Jacobi functions presented in the Preliminaries, and relation (24.5). Here we only present the proof for the ﬁrst relation since the others are similar. From relation (22.7.18) in [AbSt72]wehave 270 M.M. Rodrigues and N. Vieira

~~.m1;0/.1 22/ C m .m;0/.1 22/ m .m;0/ .1 22/ P m D P m P m 1 (24.6) 2 2 2 2 2 for m 2 0, 2 C with >m, and 0 Ä Ä 1. On the other hand, from relation (24.5) we get~~

~~. ;0/ vm m 2 Œ 2 m m P vm .1 2 / D .1/ R ./: (24.7) 2~~

Combining (24.6) and (24.7) we obtain m m Rm./ Rm1./ Rm ./; 2 D 1 C 2 2 which is equivalent to the ﬁrst relation. m Theorem 7 For N 2 , the radial function R ./ veriﬁes the following recurrence relations for distant neighborhoods: 1. Forward case

~~.2 2/.2 2/.2 / m m N C m C N C C m C N C m R ./ D C R ./ C 1 R ./I N C2N .N C C m/.2N C m/.2N C C 2/ N C2NC2~~

~~where CN and CN1 depend on the parameters , m and , and are such that~~

C0 D 1I 2. C 3/Œm2 C .1 22/. C 2/. C 4/ C1 D I . C m C 2/. m C 2/. C 4/ 2.2N C C 1/Œm2 C .2N C C 2/.1 22/.2N C / C D C 1 N .2N C C m/.2N C m/.2N C C 2/ N .2N C m/.2N C C m/.2N C 2/ C 2I .2N C C m 2/.2N C m 2/.2N C / N

~~2. Backward case~~

~~. 2 /. 2 /. 2 2 / m m C m N m N C N m R ./ D D R 2 ./ C D 1 R 2 2./I N N .2N C m 2/. C m C 2 2N/. 2n/ N N~~

~~where DN and DN1 depend on the parameters , m and , and are such that~~

~~D0 D 1I 2. 1/Œm2 C .1 22/. 2/ D1 D I . 2 m2/. 2/ 24 Fractional Circle Zernike Polynomials 271~~

~~2. C 1 2N/Œm2 C . 2N/.1 22/. C 2 2N/ D D D 1 N . m C 2 2N/. C m C 2 2N/. 2N/ N . C m C 2 2N/. m C 2 2N/. C 4 2N/ C D 2: .2N C m 4/. C m C 4 2N/. C 2 2N/ N~~

The proof will be omitted since it follows the same ideas used in the proof of the previous theorem. Finally, we introduce some differential properties of the fractional circle Zernike ˙m polynomials Z .; /. m Theorem 8 The radial function R ./ veriﬁes the following partial differential relation 2. / m 2 m C m m DR ./ Œm . m/.m .1 2 // R ./ R ./: D 2

~~Proof From relation (22.8.1) in [AbSt72]wehave~~

~~2 .m;0/ 2 2 DP m .1 2 / 2 2 2 m .1 22/ .m;0/.1 22/ m .m;0/ .1 22/: D m P m C P m 1 2 2 2 2 (24.8) On the other hand, from (24.5)wehave~~

.m;0/ 2 DP m .1 2 / 2 m 1 . 1/Œ 2 m2 m 2 m1 m 2 m R .1 2 / DR .1 2 / : D 4 C (24.9) Combining (24.5), (24.8), and (24.9), and after straightforward calculations we obtain our result. From the previous results follows immediately the next corollary. ˙m Corollary 1 The fractional circle Zernike polynomials Z .; / verify the following partial differential relation

~~2. / ˙m 2 ˙m C m ˙m DZ .; / Œm . m/.m .1 2 // Z .; / Z .; /: D~~

~~m Theorem 9 The fractional radial function R ./ satisﬁes the following partial differential equation of second order~~

~~5.1 2/ 3.32 1/ 2.2 . 2/ 2/ 2 m m C m m D R ./ DR ./ R ./ 0: 4 4 C 4 D 272 M.M. Rodrigues and N. Vieira~~

.m;0/ Proof Taking into account [MiEtAl07], the g-Jacobi function P m .x/ satisﬁes the 2 following differential equation Â Ã 2 2 .m;0/ 2 .m;0/ m C m .m;0/ .1 x / D P m .x/ Œm C .m C 2/.1 2 / DxP m .x/ C C 1 P m .x/ D 0: x 2 2 2 2 2 (24.10) On the other hand, from relation (24.5) we obtain the following equalities

~~. ;0/ m m 2 Œ 2 m m P m .1 2 / D .1/ R ./I (24.11) 2~~

Œ m h i . 1/ 2 .m;0/ 2 m2 m m1 m DP m .1 2 / D m R ./ C DR ./ I 2 4 (24.12) Œ m h . 1/ 2 2 .m;0/ 2 m4 m DP m .1 2 / D m.m C 2/ R ./ 2 16 i m3 m m1 2 m .2m C 1/ DR ./ C DR ./ : (24.13)

~~Substituting (24.11), (24.12), and (24.13)into(24.10) and after straightforward calculations, we get~~

~~5.1 2/ 3.32 1/ 2.2 . 2/ 2/ 2 m m C m m 0 D R ./ DR ./ R ./: D 4 4 C 4~~

From the previous theorem follows immediately the following corollary. ˙m Corollary 2 The fractional circle Zernike polynomials Z .; / satisfy the following partial differential equation of second order

~~5.1 2/ 3.32 1/ 2 ˙m ˙m D Z .; / DZ .; / 4 4 2.2 . C 2/ m2/ Z˙m.; / 0: C 4 D~~

~~24.4 Graphical Representation of Fractional Circle Zernike Polynomials~~

The aim of this section is to present some graphical representation for the proposed fractional circle Zernike polynomials. In the literature (see [BoWo89]) we can ﬁnd the well-known correspondences between combinations of integer values of and m and some cornea aberrations: 24 Fractional Circle Zernike Polynomials 273

~~Radial Degree . / Azimuthal Degree .m/ Cornea Aberration 0 0 Piston 1 ˙1 Tip 2 0 Defocus 2 ˙2 Astigmatism 3 ˙1 Coma 3 ˙3 Trefoil 4 0 Spherical Aberration~~

In the following three groups of ﬁgures, we present a graphical representation of fractional circle Zernike polynomials for the cases of Defocus (m D 0), Astigmatism (m D 2), and Trefoil (m D 3), for different values of including the integer case (Figures 24.1, 24.2, and 24.3).

~~Fig. 24.1 Fractional circle Zernike polynomials for Defocus considering D 2:0-plot(1,1); D 2:33-plot(1,2); D 2:66-plot(2,1); D 2:99-plot(2,2) 274 M.M. Rodrigues and N. Vieira~~

~~Fig. 24.2 Fractional circle Zernike polynomials for Astigmatism considering D 2:0-plot(1,1); D 2:33-plot(1,2); D 2:66-plot(2,1); D 2:99-plot(2,2)~~

From the three stated cases, we verify that the cornea aberration becomes more serious for values of far from the corresponding integer value already studied in the bibliography. This fact is supported by the increasing of the maximum value of the ﬁgure’s scale. The authors believe that this fact should have a medical interpretation and it could help in the study of each cornea aberration. Moreover, the analysis of the presented ﬁgures allows us to induce that this new class of fractional circle polynomials allow the possibility to establish a mathematically graduation (and therefore a exact medical graduation) for each cornea aberration. This fact will give the possibility of the development of new cornea models and the accuracy of some medical surgical procedures and techniques used in the constructions of lens. We also remark that similar graphics can be obtained for the other cornea aberrations and for another fractional circle Zernike polynomials with m 2 0 and 2 C. 24 Fractional Circle Zernike Polynomials 275

~~Fig. 24.3 Fractional circle Zernike polynomials for Trefoil considering D 3:0-plot(1,1); D 3:33-plot(1,2); D 3:66-plot(2,1); D 3:99-plot(2,2)~~

~~References~~

[AbSt72] Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. U.S. G.P.O., Washington, DC (1972) [BoWo59] Born, M., Wolf, E.: Principles of Optics. Pergamon Press, London/New York/Paris/Los Angeles (1959) [BoWo89] Born, M., Wolf, E.: The diffraction theory of aberrations. In: Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, pp. 459–490, 6th edn. Pergamon Press, New York (1989) 276 M.M. Rodrigues and N. Vieira

[IsEtAl02] Iskander, D.R., Morelande, M.R., Collins, M.J., Davis, B.: Modeling corneal surfaces with radial polynomials. IEEE Trans. Biomed. Eng. 49(4), 320–328 (2002) [KaIs06] Kasprzak, H., Iskander, D.R.: Approximating ocular surfaces by generalized conic curves. Ophthal. Physiol. Opt. 26(6), 602–609 (2006) [MiEtAl07] Mirevski, S.P., Boyadijiev, L., Scherer, R.: On the Riemann-Liouville fractional calculus, g-Jabobi functions and F-Gauss functions. Appl. Math. Comput. 187(1), 315–325 (2007) [Po99] Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Deriva- tives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999) [PrRu89] Prata, A., Rusch, W.V.T.: Algorithm for computation of Zernike polynomials expansion coefﬁcients. Appl. Opt. 28, 749–754 (1989) [PrEtAl88] Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, Volume 2: Special Functions. Gordon and Breach, New York (1988) [SaEtAl93] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993) [ScEtAl95] Schwiegerling, J., Greivenkamp, J.E., Miller, J.M.: Representation of videokerato- scopic height data with Zernike polynomials. J. Opt. Soc. Am. A 12(10), 2105–2113 (1995) Chapter 25 Double Laplace Transform and Explicit Fractional Analogue of 2D Laplacian

~~S. Rogosin and M. Dubatovskaya~~

~~25.1 Introduction~~

Fractional modeling and fractional differential equations become recently very important tool of applied mathematics (see, e.g. [GoEtAl14, KiEtAl06, SaEtAl93]). By using fractional models and solving corresponding fractional differential equation one can understand the nature of nonstandard and anomalous processes in physics, engineering, chemistry, biology, economics etc. (see, e.g., [Uc13]). One of the useful methods in the study of fractional differential equations is the integral transform method (see [DeBh14, De16]). The development of this method is an actual challenging problem. The aim of this paper is to study the properties of different type of fractional analogues of 2D Laplacian by using double integral Laplace transform. The double Laplace transform has been investigated since late 1930s. An extended description of this transform and its properties was made by D.Bernstein (her PhD thesis has been published in [Be41]). Some of the properties of the double Laplace transform are presented in Sec. 25.2. Multi-dimensional fractional differential operators are not completely recognized by fractional society. An approach describing these operators in terms of Fourier images (see, e.g., [SaEtAl93, Ch. 5]) is more theoretical than practical. Recently, two attempts were made to ﬁnd an explicit form of the fractional analog of 2D

~~S. Rogosin () • M. Dubatovskaya The Belarusian State University, Minsk, Belarus e-mail: [email protected]; [email protected]~~

~~© Springer International Publishing AG 2017 277 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_25 278 S. Rogosin and M. Dubatovskaya~~

~~Laplace operator (see [Ya10, DaYa12]), namely, these analogues have the following form:~~

~~˛;ˇ . ; / 1C˛ . ; / 1Cˇ . ; /; 0 < ˛; ˇ < 1; 4 u x y D D0C;xu x y C D0C;yu x y (25.1)~~

~~˛;ˇ . ; / ˛ ˛ . ; / ˇ ˇ . ; /; 0 < ˛; ˇ < 1: 4Cu x y DD1;xD0C;xu x y D1;yD0C;yu x y (25.2)~~

~~We consider here operator (25.1) and the skewed modiﬁcations of operator (25.2) for the unit square and for the quarter-plane, respectively (introduced in [Li11])1~~

˛;ˇIS . ; / ˛ ˇ . ; / ˛ ˇ . ; /; 0 < ˛ ˇ 2; 4C u x y DD0C;xD1;yu x y D0C;yD1;xu x y C Ä (25.3) ˛;ˇIQ . ; / ˛ ˇ . ; / ˛ ˇ . ; /; 0 < ˛ ˇ 2: 4C u x y DD0C;xD;yu x y D0C;yD;xu x y C Ä (25.4)

. /. ; / . /. ; / . /. ; / Here D0C;xu x y , D1;xu x y , D;xu x y are the left- and right-sided Riemann-Liouville fractional derivatives and the Liouville fractional derivative deﬁned for arbitrary positive and n D Œ C 1 (see, e.g., [KiEtAl06]):

Â Ã x Â Ã n Z n 1 d u.; y/d d .D u/.x; y/ D D .In u/.x; y/; 0C;x .n / dx .x / nC1 dx 0C;x 0 (25.5) Â Ã 1 Â Ã n Z n 1 d u.; y/d d .D u/.x; y/ D D .In u/.x; y/; 1;x .n / dx . x/ nC1 dx 1;x x (25.6) Â Ã ZC1 Â Ã 1 n .; / n d u y d d n .D ; u/.x; y/ D D .I ; u/.x; y/: x .n / dx . x/ nC1 dx x x (25.7) Using the double Laplace transform applied to the above said fractional analogues of the 2D laplacian we show that the proper boundary conditions for the fractional Laplace equations have the form of the so-called Cauchy type conditions. This means that the proposed fractional analogues of the 2D laplacian are correct generalizations of the standard one-dimensional fractional differential operator (see [KiEtAl06]). Main attention in this paper is paid to the operators (25.1) and (25.4). The study of another (and more cumbersome) form (25.3) is a subject of the forthcoming paper.

1One of the reasons to consider the operator in skewed form, i.e. to have both derivatives in x and y in both terms, is to enrich the set of eigenfunctions of the operator. 25 Double Laplace Transform and Fractional Laplacian 279

~~25.2 Double Laplace Transforms. Deﬁnition and Main Properties~~

~~Double Laplace transforms is deﬁned by the following relation (see, e.g., [DeBh14, p. 274]2)~~

~~ZC1 ZC1 O pxqy fO.p; q/ D .L2f / .p; q/ D .L2f .x; y// .p; q/ D f .x; y/e dxdy: 0 0 (25.8) We present here the following main properties of the double Laplace transforms following [DeBh14].~~

~~• (linearity) .L2.af C bg// .p; q/ D a .L2f / .p; q/ C b .L2g/ .p; q/; a; b 2 I • (scaling) 1 Á m n p q .L2f .a x; b y// .p; q/ D .L2f .x; y// ; ; ambn am bn~~

~~a; b >0; m; n 2 I axby • (ﬁrst shifting property) L2e f .x; y/ .p; q/ D .L2f / .p C a; q C b/I • (second shifting property)~~

~~apbq .L2f .x a; y b/H.x a/H.y b// .p; q/ D e .L2f / .p; q/; a; b >0;~~

~~H is the Heaviside function; • (transform of functions with special dependence on the arguments)~~

~~.L . // . ; / 1 Œ.L / . / .L / . / 2f x C y p q D pq f p f q I .L . // . ; / 1 Œ.L / . / .L / . / ; 2f x y p q D pCq f p C f q f is even 1 Œ.L / . / .L / . / D pCq f p f q f is oddI~~

~~• (transform of the truncation of functions)~~

~~.L . / . // . ; / 1 Œ.L / . / .L / . / 2f x H x y p q D q f p f p C q I .L . / . // . ; / 1 .L / . / 2f x H y x p q D q f p C q I .L . / . // . ; / 1 .L / . / 2f x H x C y p q D q f p I~~

~~2Similar deﬁnition with extra normalization was proposed in [DiPr65]. 280 S. Rogosin and M. Dubatovskaya~~

• (differential properties) Á @f L2 .x; y/ .p; q/ D p .L2f .x; y// .p; q/ L f .0; y/ .q/I @x Á y @f L2 .x; y/ .p; q/ D q .L2f .x; y// .p; q/ .L f .x;0// .p/I @y Á x @2f L2 .x; y/ .p; q/ D p .L2f .x; y// .p; q/ C q .L2f .x; y// .p; q/ @x@y L f .0; y/ .q/ .L f .x;0// .p/ f .0; 0/ Á y x C I @2f 2 L2 2 .x; y/ .p; q/ D p .L2f .x; y// .p; q/ p L f .0; y/ .q/ @x Á y L @f .0; / . / .0; 0/ y @x y q C f I

~~• (convolution) .L2f g/ .p; q/ D .L2f .x; y// .p; q/.L2g.x; y// .p; q/; where f g Rx Ry is the double convolution: .f g/.x; y/ D f .x t; y s/g.t; s/ dt dsI 0 0 • (inverse transform)~~

~~cZCi1 dZCi1 Á 1 1 f .x; y/ D L1fO.p; q/ .x; y/ D epxdp eqyfO.p; q/dq: 2 2 i 2 i ci1 di1 (25.9)~~

The last formula is valid if the direct double Laplace transform fO.p; q/ is an ˚analytic function in the direct product« of two semi-planes, namely in the domain .p; q/ 2 ¼2 W Re p > c; Re q > d , where c; d are some real numbers. All other formulas are valid provided existence of both of their sides. Remark 1 The double integral transform in the Laplace–Carson form (see [DiPr65])

ZC1 ZC1 N pxqy fN.p; q/ D .C2f / .p; q/ D .C2f .x; y// .p; q/ D pq f .x; y/e dxdy 0 0 (25.10) has similar properties. In particular, the scaling property has the following form (which is suitable for the fractal theory, see [NiBa13]) Á m n p q .C2f .a x; b y// .p; q/ D .C2f .x; y// ; ; am bn a; b >0; m; n 2 : Few examples illustrating the above formulas are as follows: 25 Double Laplace Transform and Fractional Laplacian 281

~~kxly Example 1 f1.x; y/ D e .~~

~~ZC1 ZC1 .pCk/x .qCl/y .L2f1/ .p; q/ D e dx e dy 0 0 Â ˇ ÃÂ ˇ Ã 1 ˇC1 1 ˇC1 D e.pCk/xˇ e.qCl/yˇ p C k xD0 q C l yD0 1 1 D : p C k q C l~~

~~Example 2 f2.x; y/ D H.x y/.~~

~~ZC1 ZC1 ZC1 Zx px qy px qy .L2f2/ .p; q/ D e dx e H.x y/dy D e dx e dy 0 0 0 0 ZC1 Â ˇ Ã 1 1 1 1 ˇC1 D epx.1 eqx/dx D epx C e.pCq/xˇ q q p p C q xD0 0 1 D : p.p C q/~~

Remark 2 There exists a problem with application of the inverse double Laplace transform. To describe this problem let us consider a simple situation, namely, when the double Laplace image is a rational function of two complex variables p and q:

~~R.p; q/ .L2f / .p; q/ D : S.p; q/~~

~~Let the denominator be represented in the form~~

Ym Yn S.p; q/ D C0 .p ak/ .q bl/; (25.11) kD1 lD1 where ak, bl are certain complex numbers (not necessarily different). Then to recover an initial function f .x; y/ one needs to apply Residue Theorem in two semi-planes of variables p and q, respectively. But, if the polynomial S.p; q/ is irreducible, i.e. there exist no nontrivial polynomials S1.p; q/, S2.p; q/ of the above described form (25.11), such that S.p; q/ D S1.p; q/S2.p; q/, then such technique cannot be applied. The simplest example of the irreducible polynomial is S.p; q/ D p2 C q2 1. Let us repeat a simple example of application of the double Laplace transform to solving of an initial-boundary problem for a second order differential equation in the quarter-plane (see [DeBh14, p. 278]). 282 S. Rogosin and M. Dubatovskaya

~~Example 3 Let us consider the wave equation 2 c uxx D utt; x 0; t >0; (25.12) under the following initial conditions~~

~~u.x;0/D f .x/; ut.x;0/D g.x/; (25.13) and the following boundary conditions~~

u.0; t/ D 0; ux.0; t/ D 0: (25.14) Assume that the given functions f .x/ and g.x/ are suitable to apply the below technique (e.g. are absolutely integrable in x 2 C). Let us apply the double Laplace transform uO.p; q/ D .L2u.x; t// .p; q/ to the equation (25.12): 2 c .L2uxx.x; t// .p; q/ D .L2utt.x; t// .p; q/: By using the differential properties of the double Laplace transform we have h i 2 2 O c p uO.p; q/ p .L2u.0; t// .q/ .L2ux.0; t// .q/ C u.0; 0/ h i 2 O D p uO.p; q/ q .L2u.x;0// .p/ .L2ut.x;0// .p/ C u.0; 0/ :

Applying conditions (25.13), (25.14) we arrive at the following relation h i c2p2 q2 uO.p; q/ D qfO.p/ COg.p/ or qfO.p/ COg.p/ uO.p; q/ D : (25.15) q2 c2p2 The direct calculation shows that with a > cRe p we have

~~aZCi1 1 qfO.p/ COg.p/ 1 eqtdq D fO.p/ cosh cpt C gO.p/ sinh cpt: 2 i q2 c2p2 cp ai1 Thus the inverse double Laplace transform (25.9) applied to (25.15)givesforany b >0~~

bZCi1Ä 1 1 u.x; t/ D fO.p/ cosh cpt C gO.p/ sinh cpt epxdp 2 i cp bi1 2 3 Zxct xZCct 1 1 D Œf .x C ct/ C f .x ct/ C 4 g./d g./d5 : 2 2c 0 0 25 Double Laplace Transform and Fractional Laplacian 283

Therefore, the solution to the initial-boundary value problem (25.13)–(25.14)for equation (25.12) has the following form (known in the literature d’Alembert formula for the solution of 2D wave equation) 2 3 xZCct 1 1 u.x; t/ D Œf .x C ct/ C f .x ct/ C 4 g./d5 : (25.16) 2 2c xct

~~25.3 Application of the Double Laplace Transform to Explicit Fractional Laplacian~~

~~In this section we present the values of the double Laplace transform of the fractional analogues (25.1) and (25.4) of the 2D laplacian.~~

~~25.3.1 Power Type Fractional Laplacian 4˛;ˇ~~

Let us apply the double Laplace transform to the fractional laplacian (25.1). It is L 1C˛ . ; / . ; / 0<˛<1 sufﬁcient to calculate 2D0C;xu x y p q with : 0 1 ZC1 ZC1 Zx 2 1 . ; / 1C˛ qy px @ d u t y dt A L2D u.x; y/ .p; q/ D e dy e dx: 0C;x dx2 .1 ˛/ .x t/˛ 0 0 0 (25.17) Denote for shortness

~~Zx 1 . ; / u t y dt 1˛ '.x; y/ WD D I0 ; u.x; y/: .1 ˛/ .x t/˛ C x 0~~

Then by applying the differentiation property of the Laplace transform to the internal integral in (25.17)wehave Â Ã d2 d' L '.x; y/ .p/ D p2 .L '.x; y// .p/ p'.0;y/ .0; y/ x dx2 x dx 2 .L '. ; // . / . 1˛ . ; //.0/ . ˛ . ; //.0/: D p x x y p p I0C;xu x y D0C;xu x y

~~Further, we use the formula for the Laplace transform of the fractional integral [SaEtAl93, p. 140] 284 S. Rogosin and M. Dubatovskaya L 1˛ 1C˛ .L / : xI0C;x D p x (25.18)~~

Hence 0 1 ZC1 Zx d2 1 u.t; y/dt L D1C˛ u.x; y/ .p/ D epx @ A dx x 0C;x dx2 .1 ˛/ .x t/˛ 0 0 (25.19) 1C˛ .L . ; // . / . 1˛ . ; //.0/ . ˛ . ; //.0/: D p xu x y p p I0C;xu x y D0C;xu x y Applying the external integral in (25.17) and using the symmetry of the formula (25.1) we obtain the following result. Proposition 1 The double Laplace transform of the fractional laplacian (25.1)is represented in the following form ˛;ˇ 1C˛ 1Cˇ L24 u.x; y/ .p; q/ D p C q .L2u.x; y// .p; q/ Â Ã Â Ã Á L 1˛ . ; / ˇ . / L 1ˇ . ; / ˇ . / p y I0C;xu x y ˇ q q x I0C;yu x y ˇ p xD0 yD0 (25.20) Â Ã Â Ã Á L ˛ . ; / ˇ . / L ˇ . ; / ˇ . /: y D0C;xu x y ˇ q x D0C;yu x y ˇ p xD0 yD0

Remark 3 It follows from [SaEtAl93, Thm. 3.1, Thm. 3.6] that if the function u is either Hölder continuous in both variables, or Lr-integrable on C C with sufﬁciently large r, then both terms in the second line of (25.20) vanish.

~~˛;ˇIQ 25.3.2 Skewed Fractional Laplacian 4C in the Quarter-Plane~~

In this subsection we determine the double Laplace transform of the skewed fractional laplacian in the quarter-plane (25.4). It is natural (see [DaYa12]) to consider such construction in three cases: (a) 0<˛;ˇ<1; (b) 0<˛<1;1<ˇ<2I ˛ C ˇ D 2; (c) 1<˛<2;0<ˇ<1I ˛ C ˇ D 2. ˛;ˇIQ Case (a). Due to the symmetry of the operator 4C it suffices to transform the first term in (25.4): . ; / ˛ ˇ : d1 x y WD D0C;xD;y Note as in previous subsection that for any 0< <1and sufficiently smooth 25 Double Laplace Transform and Fractional Laplacian 285

~~Zz 1 d ./d d 1 .D /.z/ D D .I /.z/; (25.21) 0C;z .1 / dz .z / dz 0C;z 0~~

~~ZC1 1 ./ d d d 1 .D ; /.z/ D D .I ; /.z/: (25.22) z .1 / dz . z/ dz z z~~

~~1 Besides, the right-sided fractional integral I;z is related to the left-sided fractional integral by two formulas (see [SaEtAl93, formulas (11.27), (11.29)])~~

~~. /. / . /. / . /. /; I;z z D cos I0C;z z C sin I0C;zx St z (25.23)~~

~~. /. / . /. / . /. /; I;z z D cos I0C;z z C sin SI0C;x z (25.24) where S is the singular integral operator over semi-axis and compositions are deﬁned as follows:~~

~~z C1 Z Z 1 dx 1 t .t/dt .I x St /.z/ D ; 0C;z . / .z x/1 x t x 0 0~~

~~ZC1 Zx 1 dx 1 .t/dt .SI /.z/ D : 0C;x .x z/ . / .x t/1 0 0~~

~~It follows from (25.21), (25.22), (25.23) that the operator d1.x; y/ possesses the following representation~~

~~d 1˛ d 1ˇ d1.x; y/ D I I dx 0C;x dy ;y h i (25.25) d d 1 ˇ 1 ˇ D I1˛ cos ˇ I sin ˇ I !.t;/ ; dx 0C;x dy 0C;y 0C;y where~~

~~!.x; y/ WD y1Cˇ.Sr1ˇu.x; r//.y/: (25.26)~~

~~Denoting~~

d 1ˇ '1.x; y/ WD .I u.x; t//.y/ dy 0C;y we get by using Laplace transform properties 286 S. Rogosin and M. Dubatovskaya Â Ã d 1˛ L2 I '1.x; y/ .p; q/ dx 0C;x L . ˛L ' . ; // . / . 1˛ ' . ; //.0/ . / D y p x 1 x y p I0C;x 1 t y q Â Â Ã Ã ˛ d 1ˇ 1˛ D p L L .I u.x; //.y/ .q/ .p/ L .I '1.t; y//.0/ .q/ x y dy 0C;y y 0C;x Á ˛ ˇ .L . ; // . ; / ˛ L . 1ˇ . ; //.0/ . / D p q 2u x y p q p x I0C;yu x t p Â Ã d 1ˇ L .I '2.0; t//.y/ .q/; y dy 0C;y where ' .0; / . 1˛ .; //.0/: 2 t D I0C;x u t Since Â Ã d 1ˇ ˇ 1ˇ L .I '2.0; t//.y/ .q/ D q L '2.0; y/ .q/ .I '2.0; t//.0/; y dy 0C;y y 0C;y and . 1ˇ ' .0; //.0/ I0C;y 2 0 1 0 1 Zy Zx B C 1 1 B d @ u.t;/dt A C c˛;ˇ ; D .1 ˇ/ .1 ˛/ @ . /ˇ . /˛ ˇ A D Iu y x t ˇ ˇ 0 0 ˇ ˇ xD0 ˇ yD0 then Â Ã d 1˛ d 1ˇ L2 I .I u.x; //.y/ .p; q/ dx 0C;x dy 0C;y Â Â ÃÃ ˛ ˇ ˛ 1ˇ ˇ D p q .L2u.x; y// .p; q/ p L I u.x;/ˇ .p/ (25.27) x 0C;y yD0 Â Â ÃÃ ˇ 1˛ ˇ q L I0 ; u.t; y/ˇ .q/ c˛;ˇ : y C x xD0 Iu

Similar to the previous calculation we have Â Ã d 1˛ d 1ˇ L2 I .I !.x; //.y/ .p; q/ dx 0C;x dy 0C;y Â Â ÃÃ ˛ ˇ ˛ 1ˇ ˇ D p q .L2!.x; y// .p; q/ p L I !.x;/ˇ .p/ (25.28) x 0C;y yD0 Â Â ÃÃ ˇ 1˛ ˇ q L I0 ; !.t; y/ˇ .q/ c˛;ˇ ! ; y C x xD0 I 25 Double Laplace Transform and Fractional Laplacian 287 where 0 1 0 1 Zy Zx B C 1 1 B dt @ !.t;/dt A C c˛;ˇ ! : I D .1 ˇ/ .1 ˛/ @ . /ˇ . /˛ ˇ A y x t ˇ ˇ 0 0 ˇ ˇ xD0 ˇ yD0

~~˛;ˇIQ It follows from (25.23) that the operator 4C has the representation:~~

˛;ˇIQ . ; / 4C u x y ÄÂ Ã Â Ã d d 1 ˇ d d 1 ˇ D cos ˇ I1˛ I u.x; y/ C I1˛ I u.x; y/ dx 0C;x dy 0C;y dy 0C;y dx 0C;x ÄÂ Ã Â Ã d d 1 ˇ d d 1 ˇ sin ˇ I1˛ I !.x; y/ C I1˛ I !.x; y/ dx 0C;x dy 0C;y dy 0C;y dx 0C;x (25.29) where !.x; y/ is deﬁned in (25.26). Then, by (25.27) and (25.28) we get the following result. Proposition 2 Let 0<˛;ˇ<1. Let !.x; y/ be deﬁned in (25.26). Then the double Laplace transform of the skewed fractional laplacian (25.4) is represented in the following form Á L ˛;ˇIQ . ; / . ; / 24C u x y p q ˛ ˇ ˇ ˛ D p q C p q .L2 .cos ˇu.x; y/ sin ˇ!.x; y/// .p; q/ Â Á Ã ˛ L 1ˇ . ˇ . ;/ ˇ!. ;// ˇ . / p x I0C;y cos u x sin x ˇ p yD0 Â Ã ˇ 1˛ ˇ p L I0 ; .cos ˇu.x;/ sin ˇ!.x;// ˇ .p/ x C y yD0 (25.30) Â Á Ã ˛ L 1ˇ . ˇ . ; / ˇ!. ; // ˇ . / q y I0C;x cos u t y sin t y ˇ q xD0 Â Ã ˇ 1˛ ˇ q L I0 ; .cos ˇu.t; y/ sin ˇ!.t; y// ˇ .p/ x C x xD0

c˛;ˇIu c˛;ˇI! cˇ;˛Iu cˇ;˛I! ; where constants c˛;ˇIu; c˛;ˇI! are deﬁned in (25.27) and (25.28), respectively, and constants cˇ;˛Iu; cˇ;˛I! are deﬁned similarly by interchanging parameters in fractional integrals in (25.27) and (25.28). 288 S. Rogosin and M. Dubatovskaya

Case (b). Here we consider the double Laplace transform of the skewed ˛;ˇIQ 0<˛<1;1<ˇ<2 fractional laplacian 4C with parameters . The remaining case (c) can be obtained by interchanging ˛ and ˇ. As before it sufﬁces to apply the double Laplace transform only to the ﬁrst term ˛;ˇIQ of 4C :

~~Â 2 Ã ˛ ˇ d 1˛ d 2ˇ dQ1.x; y/ DD D u.x; y/ D I I u.t; y/ .x; y/: 0C;x ;y dx 0C;x dy2 ;y~~

~~Using the relation between right- and left-sided fractional integrals (25.23)we represent dQ1.x; y/ in the form~~

dQ1.x; y/ Â 2 h iÃ d d 2 ˇ 2 ˇ D I1˛ cos ˇI u.t;/ sin ˇI ˇ2Sr2ˇu.t; r/ .x; y/ dx 0C;x dy2 0C;y 0C;y Â 2 Ã d d 2 ˇ Dcos ˇ I1˛ I u.t;/ .x; y/ dx 0C;x dy2 0C;y Â 2 Ã d d 2 ˇ C sin ˇ I1˛ I !.Q t;/ .x; y/ dx 0C;x dy2 0C;y

~~DW dQ11.x; y/ C dQ12.x; y/: where~~

!.Q t;/D ˇ2Sr2ˇu.r;/: (25.31) Â Â Ã Ã 2 d 1˛ d 2ˇ L2dQ11.x; y/ .p; q/ Dcos ˇL L I I u.t;/ .x; y/ .p; q/ y x dx 0C;x dy2 0C;y Â Â Ã Ã d 1˛ Dcos ˇL L I 'Q1.t; y/ .x; y/ .p; q/; y x dx 0C;x where

~~Â 2 Ã d 2ˇ 'Q1.; y/ D I u.t;/ .y/: dy2 0C;y~~

~~Hence L2dQ11.x; y/ .p; q/ Â Â ÃÃ (25.32) ˛ 1˛ ˇ Dcos ˇL p .L 'Q1.x; y//.p/ I0 ; 'Q1.t; y/ˇ .q/: y x C x xD0 25 Double Laplace Transform and Fractional Laplacian 289~~

~~If the function u.x; y/ is such that the operators Ly and Lx are interchangeable, then~~

˛ Ly .p .Lx'Q1.x; y//.p// .q/ ˛ D p .LxLy .'Q1.x; y// .p; q/ Â Â 2 ÃÃ d 2 ˇ D p˛ L L I u.t;/ .p; q/ x y dy2 0C;y ˛ ˇ D p q .L2u.x; y// .p; q/ Â Á Á Ã ˛ L 2ˇ . ;/ ˇ L ˇ1 . ;/ ˇ . /: p q x I0C;yu x ˇ C x D0C;yu x ˇ p yD0 yD0

Besides, if the fractional integrals in the second term in (25.32), then Â Ã 1˛ ˇ L I0 ; 'Q1.t; y/ˇ .q/ y C x xD0 Â Ã 2 d 2ˇ 1˛ ˇ D L I I0 ; u.t;/ ˇ .q/ y dy2 0C;y C x xD0 Â Ã Â Ã ˇ 1˛ ˇ 2ˇ 1˛ ˇ Dq L I0 ; u.t; y/ ˇ .q/ C q I I0 ; u.t;/ ˇ ˇ y C x xD0 0C;y C x xD0 ˇ yD0 Â Ã ˇ1 1˛ ˇ C D I0 ; u.t;/ ˇ ˇ : 0C;y C x xD0 ˇ yD0

~~Analogously, Â Â Ã Ã 2 d 1˛ d 2ˇ L2dQ12.x; y/ .p; q/ D sin ˇL L I I !.Q t;/ .x; y/ .p; q/: y x dx 0C;x dy2 0C;y~~

~~Under similar conditions as before we have Â Â ÃÃ ˛ 1˛ ˇ L2dQ12.x; y/ .p; q/ D sin ˇL p .L 'Q2.x; y//.p/ I0 ; 'Q2.t; y/ˇ .q/; y x C x xD0 (25.33) where~~

~~Â 2 Ã d 2ˇ 'Q2.; y/ D I !.Q t;/ .y/: dy2 0C;y 290 S. Rogosin and M. Dubatovskaya~~

If the function u.x; y/ is such that LyLx'Q2.x; y/ D LxLy'Q2.x; y/, then ˛ Ly .p .Lx'Q2.x; y//.p// .q/ ˛ D p .LxLy .'Q2.x; y// .p; q/ ˛ ˇ D p q .L2!.Q x; y// .p; q/ Â Á Á Ã ˛ L 2ˇ !. ;/ ˇ L ˇ1 !. ;/ ˇ . /: p q x I0C;y Q x ˇ C x D0C;y Q x ˇ p yD0 yD0

Besides, if the fractional integrals in the second term in (25.33), then Â Ã 1˛ ˇ L I0 ; 'Q2.t; y/ˇ .q/ y C x xD0 Â Ã 2 d 2ˇ 1˛ ˇ D L I I0 ; !.Q t;/ ˇ .q/ y dy2 0C;y C x xD0 Â Ã Â Ã ˇ 1˛ ˇ 2ˇ 1˛ ˇ Dq L I0 ; !.Q t; y/ ˇ .q/ C q I I0 ; !.Q t;/ ˇ ˇ y C x xD0 0C;y C x xD0 ˇ yD0 Â Ã ˇ1 1˛ ˇ C D I0 ; !.Q t;/ ˇ ˇ : 0C;y C x xD0 ˇ yD0

~~˛;ˇIQ The results for the second term of 4C :~~

Â 2 Ã ˛ ˇ d 1˛ d 2ˇ dQ2.x; y/ DD D u.x; y/ D I I u.t;/ .x; y/ 0C;y ;x dy 0C;y dx2 ;x Â 2 Ã d d 2 ˇ Dcos ˇ I1˛ I u.t;/ .x; y/ dy 0C;y dx2 0C;x Â 2 Ã d d 2 ˇ C sin ˇ I1˛ I !.Q t;/ .x; y/ dy 0C;y dx2 0C;x

DW dQ21.x; y/ C dQ22.x; y/ can be obtained simply by interchanging variables x and y. Combining the above results we obtain the following: Proposition 3 Let 0<˛<1;1<ˇ<2. Let the above conditions be satisﬁed. Let

.x; y/ WD cos ˇu.x; y/ C sin ˇ!.Q x; y/; where !Q is deﬁned in (25.31). Then the double Laplace transform of the skewed ˛;ˇIQ fractional laplacian 4C has the following form 25 Double Laplace Transform and Fractional Laplacian 291 Á L ˛;ˇIQ . ; / . ; / 24C u x y p q

˛ ˇ ˇ ˛ D .p q C p q / .L2.x; y// .p; q/ Â Á Ã Â Á Ã ˛ L 2ˇ . ;/ ˇ . / ˛ L 2ˇ . ; / ˇ . / p q x I0C;y x ˇ p q p y I0C;x t y ˇ q yD0 xD0 Â Á Ã Â Á Ã ˛ L ˇ1 . ;/ ˇ . / ˛ L ˇ1 . ; / ˇ . / p x D0C;y x ˇ p q y D0C;x t y ˇ q yD0 xD0 Â Ã Â Ã ˇ 1˛ ˇ ˇ 1˛ ˇ q L I0 ; .t; y/ ˇ .q/ p L I0 ; .x;/ ˇ .p/ y C x xD0 x C y yD0 Â Ã Â Ã 2ˇ 1˛ ˇ 2ˇ 1˛ ˇ q I I0 ; .t;/ ˇ ˇ C p I I0 ; .t;/ ˇ ˇ 0C;y C x xD0 ˇ 0C;x C y yD0 ˇ yD0 xD0 Â Ã Â Ã ˇ1 1˛ ˇ ˇ1 1˛ ˇ C D I0 ; .t;/ ˇ ˇ C D I0 ; .t;/ ˇ ˇ : 0C;y C x xD0 ˇ 0C;x C y yD0 ˇ yD0 xD0 (25.34) Remark 4 If the function u.x; y/ is smooth enough (e.g. Hölder continuous in both variables), then all terms containing values of fractional integrals at the initial point vanish. In this case formula (25.34) contains only terms with double Laplace transform (1st line) and those with Laplace transform of the fractional derivative (3rd line). Analogously, in this case formula (25.30) contains only term with double Laplace transform (1st line).

~~25.4 Discussion and Outlook~~

In this paper we apply the double Laplace transform to some explicit fractional analogues of the 2D laplacian. Using the obtained results we propose to use the Cauchy type boundary conditions for considered fractional Laplace equation. These conditions differ from the standard conditions for the Laplace equations and they have the nonlocal nature. Thus, such problems describe the processes with nonlocal behaviour. In the forthcoming paper we suppose to fix proper functional spaces and to find corresponding solvability conditions of the Cauchy type problem for fractional Laplace equation as well as to determine the set of eigenfunctions and the fundamental solution. Since it is known (see [SaEtAl93]) that there exists no unique fractional analog of the Laplace operator, then the question remains to find additional properties of the proposed explicit fractional analogues of the Laplace operator. 292 S. Rogosin and M. Dubatovskaya

Acknowledgements The work is partially supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007–2013/ under REA grant agreement PIRSES-GA-2013-610547 - TAMER and by Belarusian Fund for Fundamental Scientiﬁc Research (grant F17MS-002). The authors are grateful to an anonymous referee for attentive reading of the paper and making important remarks improving the presentation.

~~References~~

[Be41] Bernstein, D.L.: The double Laplace integral. Duke Math. J. 8(3), 460–496 (1941) [DaYa12] Dalla Riva, M., Yakubovich, S.: On a Riemann–Liouville fractional analog of the Laplace operator with positive energy. Integr. Transf. Spec. Funct. 23(4), 277–295 (2012) [De16] Debnath, L.: The double Laplace transforms and their properties with applications to functional, integral and partial differential equations. Int. J. Appl. Comput. Math. 2(2), 223–241 (2016) [DeBh14] Debnath, L., Bhatta, D.: Integral Transforms and Their Applications, 3rd edn. CRC Press, Boca Raton/New York (2014) [DiPr65] Ditkin, V.A., Prudnikov, A.P.: Integral Transforms and Operational Calculus. Perga- mon Press, Oxford (1965) [GoEtAl14] Gorenﬂo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Lefﬂer Functions, Related Topics and Applications. Springer, Berlin (2014) [KiEtAl06] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) [Li11] Lipnevich, V.V.: Fractional analog of the Laplace operator and its simplest properties. Proc. Inst. Math. Minsk 19(2), 82–86 (2011) [in Russian] [NiBa13] Nigmatullin, R.R., Baleanu, D.: New relationships connecting a class of fractal objects and fractional integrals in space. Fract. Calc. Appl. Anal. 16(4), 911–936 (2013) [SaEtAl93] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993) [Uc13] Uchaikin, V.V.: Fractional Derivatives for Physicists and Engineers, I and II. Speringer, Berlin; Higher Educational Press, Beijing (2013) [Ya10] Yakubovich, S.: Eigenfunctions and fundamental solutions of the two-parameters laplacian. Int. J. Math. Math. Sci. 2010, Article ID 541934, 18 pp. (2010) Chapter 26 Stability of the Laplace Single Layer Boundary Integral Operator in Sobolev Spaces

~~O. Steinbach~~

~~26.1 Introduction~~

~~As a model problem we consider the Dirichlet boundary value problem for the Laplace equation,~~

u.x/ D 0 for x 2 ˝; u.x/ D g.x/ for x 2 D @˝; (26.1) where ˝ 3 is a bounded Lipschitz polyhedron. Using an indirect approach, the solution of (26.1) can be described as single layer potential Z e u.x/ D .Vw/.x/ WD U .x; y/w.y/ dsy for x 2 ˝; (26.2) where 1 1 U.x; y/ D 4 jx yj is the fundamental solution of the Laplacian. It is well known, see, e.g., [Co88] that eV W H1=2. / ! H1.˝/. The unknown density w 2 H1=2. / is then found by int 1 1=2 applying the interior Dirichlet trace operator 0 W H .˝/ ! H . / to (26.2) which results in the boundary integral equation Z .Vw/.x/ WD U .x; y/w.y/ dsy D g.x/ for x 2 ; (26.3)

~~O. Steinbach () TU Graz, Graz, Austria e-mail: [email protected]~~

© Springer International Publishing AG 2017 293 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_26 294 O. Steinbach and which is equivalent to a Galerkin–Bubnov formulation: Find w 2 H1=2. / such that

~~1=2 hVw;vi Dhg;vi for all v 2 H . /: (26.4)~~

~~inte 1=2 Since the single layer boundary integral operator V D 0 V W H . / ! H1=2. / is elliptic, [HsWe77],~~

V 2 1=2 ; . /; hVw wi c1 kwkH1=2. / for all w 2 H (26.5) unique solvability of the variational formulation (26.4) follows. Moreover we can deduce a stability and error analysis of related boundary element discretization schemes, see, e.g., [St08]. Error estimates then rely on the regularity of w D V1g, i.e. on the regularity of the given Dirichlet datum g, and on the mapping properties of the single layer boundary integral operator V. In the case of a Lipschitz domain ˝ 1=2Cs. / 1=2Cs. / Œ 1 ; 1 we have that V W H ! H is bijective for all s 2 2 2 ,see [Co88, Ve84], while in the case of a polyhedral bounded domain this remains true < > 1 for jsj s0 where s0 2 is determined by the related interior and exterior angles in corners and at edges, see, e.g., [Gr92] and [CoSt85] for the two-dimensional case. The error estimate for the Galerkin solution of the Galerkin–Bubnov variational formulation (26.4) is given, due to Cea’s lemma, in the energy norm in H1=2. /. Hence, to derive error estimates in stronger norms, e.g. in L2. /,wehavetouse an inverse inequality for the used boundary element space and where we have to assume a globally quasi-uniform boundary element mesh, see, e.g., [St08]fora more detailed discussion. In fact, this excludes non-uniform and adaptive meshes as often used in practice. Instead of the Galerkin–Bubnov variational formulation (26.4) we will consider a Galerkin–Petrov variational formulation which allows the use of different trial and test spaces, both in the continuous and discrete setting. In this case, the ellipticity estimate (26.5) has to be replaced by an appropriate stability condition, also known as inf sup condition. While the analysis of the Galerkin–Bubnov formulation (26.4) relies on a related domain variational formulation in H1.˝/, our analysis is based on using a Galerkin–Petrov domain variational formulation for which we have to introduce suitable Sobolev spaces. With this we can not only conclude known mapping properties of the single layer boundary integral operator, but we can also establish a new stability condition which ensures unique solvability of the Galerkin– Petrov variational formulation. In this note we will not consider a stability and error analysis of related Galerkin– Petrov boundary element methods which will be a topic for further research. In fact, such an approach can also be used for Galerkin–Petrov variational formulations in weaker Sobolev spaces, e.g., when the given Dirichlet data have reduced regularity, for example if we have g 2 L2. / only, see, e.g., [ApEtAl16]. However, the main focus of future work will be on the extension of this concept to the mathematical and numerical analysis of boundary integral equation and boundary element methods for time-dependent problems such as the heat equation, see, e.g., [Co90] for related Galerkin–Bubnov formulations. 26 Stability of the Laplace Single Layer Boundary Integral Operator 295

~~26.2 Strong Domain Variational Formulation~~

For the Dirichlet boundary value problem (26.1) we consider, instead of a standard domain variational formulation in H1.˝/ which is based on Green’s ﬁrst formula, a Galerkin–Petrov variational formulation. For this we introduce n o 1 2 1 H.˝/ WD v 2 H .˝/ W v 2 L .˝/ H .˝/; with the norm v 2 v 2 v 2 v 2 : k kH.˝/ Dk kL2.˝/ Ckr kL2.˝/ Ck kL2.˝/

Then we have to ﬁnd u 2 H.˝/ satisfying u.x/ D g.x/ for x 2 such that Z Œu.x/v.x/ dx D 0 for all v 2 L2.˝/; (26.6) ˝ where we have to assume that the given Dirichlet datum g is in the Dirichlet trace int 1=2 space 0 H.˝/ H . /. In particular, let ug 2 H.˝/ be a bounded and norm int preserving extension of g 2 0 H.˝/ with

~~int v .˝/ .˝/ : kgk 0 H.˝/ D min k kH DkugkH (26.7) v2H.˝/Wvj Dg~~

1 It remains to ﬁnd u0 2 XS WD H.˝/ \ H0 .˝/ such that Z Z 2 aS.u0;v/WD Œu0.x/v.x/ dx D Œug.x/v.x/ dx for all v 2 YS WD L .˝/: ˝ ˝ (26.8) Related to the trial and test spaces we introduce the associated norms

~~h i1=2 2 2 ; v v 2 : kukXS WD krukL2.˝/ Ck ukL2.˝/ k kYS WD k kL .˝/~~

~~Lemma 1 The bilinear form of the variational problem (26.8) is bounded, i.e.~~

. ;v/ v ;v ; jaS u jÄkukXS k kYS for all u 2 XS 2 YS and satisﬁes the stability condition s a .u;v/ .˝/ c kuk Ä sup S for all u 2 X ; c D min S XS v S S 1 .˝/ 0¤v2YS k kYS C min where min.˝/ is the minimal eigenvalue of the Dirichlet eigenvalue problem

~~u.x/ D u.x/ for x 2 ˝; u.x/ D 0 for x 2 : 296 O. Steinbach~~

Proof The boundedness of the bilinear form aS.; / is a direct consequence of the Cauchy–Schwarz inequality, ˇZ ˇ ˇ ˇ . ;v/ ˇ Œ . /v. / ˇ 2 v 2 v : jaS u jDˇ u x x dxˇ Äk ukL .˝/k kL .˝/ ÄkukXS k kYS ˝

To prove the stability condition we consider u 2 XS and choose v D u u 2 YS. By using the minimal Dirichlet eigenvalue for the Laplacian in ˝, 2 krvk 2 .˝/ L .˝/ ; min D min 2 0 v 1.˝/ v ¤ 2H0 k kL2.˝/ and Hölders inequality we have

~~v 2 2 2 k kYS Dku ukL .˝/ ÄkukL .˝/ Ck ukL .˝/ 1 p Ä krukL2.˝/ Ck ukL2.˝/ min.˝/ s 1 Á1=2 Á1=2 1 .˝/ 1 2 2 C min : Ä C krukL2.˝/ Ck ukL2.˝/ D kukXS min.˝/ min.˝/ Then, Z~~

~~aS.u;v/D aS.u; u u/ D Œu.x/ Œu.x/ u.x/ dx ˝ Z Z D Œu.x/u.x/ dx C Œu.x/2 dx ˝ ˝ Z Z D jru.x/j2 dx C Œu.x/2 dx ˝ ˝ s~~

2 2 2 min.˝/ u 2 u 2 u u v DkrkL .˝/ Ck kL .˝/ Dk kXS k kXS k kYS 1 C min.˝/ implies the stability condition as claimed. ut As a consequence of Lemma 1 we conclude unique solvability of the variational problem (26.8) to obtain u D u0 C ug 2 H.˝/. In particular, when choosing in (26.6) v Du 2 L2.˝/, this gives Z 2 Œ . /2 0: k ukL2.˝/ D u x dx D (26.9) ˝

1 For the solution u 2 H.˝/ H .˝/ of the variational formulation (26.6) we note that the interior Neumann trace @ int 1 u.x/ WD lim nx rxQu.xQ/ D u.x/ for x 2 (26.10) ˝3Qx!x2 @nx 26 Stability of the Laplace Single Layer Boundary Integral Operator 297

int 1=2 is well defined, at least we have 1 u 2 H . / due to duality arguments and the use of Green’s first formula. To do a more detailed analysis, for u 2 H.˝/ we define D u 2 L2.˝/ and we consider the Dirichlet boundary value problem

~~.x/ D .x/ for x 2 ˝; .x/ D 0 for x 2 :~~

In the case of a domain ˝ with a sufficient smooth boundary or in the case of a 2 2 convex polyhedron we find 2 H .˝/, and therefore H.˝/ D H .˝/ follows. However, this is not true when the domain ˝ is polyhedral bounded, but non- 2 convex. In this case, H.˝/ includes harmonic functions which are not in H .˝/ s.˝/ < > 3 but in H , s si,forsomesi 2 , see, for example, [Gr92, Corollary 2.6.7]. In int int any case, the Neumann trace operator 1 W H.˝/ ! 1 H.˝/ is well defined, int implying the Neumann trace space 1 H.˝/, and satisfying int v int v .˝/ v .˝/: k 1 k 1 H. / Ä cN k kH for all 2 H (26.11)

~~Lemma 2 Let u 2 H.˝/ be the unique solution of the variational formulation (26.6). Then, int 2 2 : ci k 1 uk int Äkruk 2 (26.12) 1 H.˝/ L .˝/~~

Proof Let u D u0 C ug 2 H.˝/ be the unique solution of (26.8). We then deﬁne Z Z 1 eu.x/ D u.x/ ; D g.x/ dsx; eu.x/ dsx D 0; j j and whereeu is the unique solution of the Dirichlet boundary value problem

~~eu.x/ D 0 for x 2 ˝; eu.x/ D g.x/ for x 2 :~~

Obviously,eu 2 H.˝/, and (26.11) together with (26.9) then implies 1 int 2 2 2 2 2 k euk int Äkeuk Dkeuk 2 Ckreuk 2 Ckeuk 2 1 H.˝/ H.˝/ L .˝/ L .˝/ L .˝/ cN 1 e 2 e 2 e 2 : DkukL2.˝/ CkrukL2.˝/ DkukH1.˝/ Since an equivalent norm in H1.˝/ is given by

~~ÄZ 2 v 2 v. / v 2 ; k k 1 .˝/ WD x dsx Ckr kL2.˝/ H we immediately conclude 1 int 2 2 2 eu c eu 1 c eu 2 : k 1 k 1H.˝/ Ä k kH .˝/ D kr kL .˝/ cN ~~

~~Insertingeu D u concludes the proof. ut 298 O. Steinbach~~

~~In addition to the interior Dirichlet boundary value problem (26.1)wealso consider the exterior Dirichlet problem~~

~~u.x/ D 0 for x 2 ˝c WD nn˝; u.x/ D g.x/ for x 2 ; (26.13) where in addition we have to impose a suitable radiation condition,~~

~~u.x/ D O.1=jxj/ as jxj!1: (26.14)~~

3 When introducing the bounded domain ˝r WD Brn˝ with Br WD fx 2 Wjxj < rg, and choosing r >0such that ˝ Br, we can proceed as in the case of the interior Dirichlet boundary value problem (26.1), when considering the limit r !1and the radiation condition (26.14). As in (26.12) we may deﬁne the exterior Neumann trace of the solution u satisfying ext 2 2 c k 1 uk ext c Äkruk 2 c : (26.15) e 1 H.˝ / L .˝ / .˝c/ s.˝c/ < > 3 Note that H Â H , s se,forsomese 2 which may differ from si.

~~26.3 Ultra-Weak Domain Variational Formulation~~

To derive mapping properties of the single layer boundary integral operator V we may also consider the ultra-weak domain variational formulation, see, e.g., [ApEtAl16]. Multiplying the partial differential equation in (26.1)withatest 1 function v 2 H.˝/ \ H0 .˝/, integrating over ˝, and applying integration by parts twice, this gives Z Z 0 D Œu.x/v.x/ dx D ru.x/ rv.x/ dx ˝ ˝ Z Z @ D u.x/Œv.x/ dx C u.x/ v.x/ dsx : ˝ @nx

When inserting the Dirichlet boundary condition, this results in the Galerkin–Petrov 2 variational formulation to ﬁnd u 2 XU D L .˝/ such that Z Z @ aU.u;v/WD u.x/Œv.x/ dx D g.x/ v.x/ dsx (26.16) ˝ @nx

1 is satisﬁed for all v 2 YU D H.˝/ \ H0 .˝/. In this case we have to assume that the given Dirichlet datum g is in the dual of the interior Neumann trace space, i.e. int 0 g 2 Œ 1 H.˝/ . We obviously have XU D YS and YU D XS, respectively, with the associated norms

~~h i1=2 2 2 2 ; v v v : kukXU DkukL .˝/ k kYU D kr kL2.˝/ Ck kL2.˝/ 26 Stability of the Laplace Single Layer Boundary Integral Operator 299~~

~~SimilarasinLemma1 we can prove boundedness, . ;v/ v ;v ; jaU u jÄkukXU k kYU for all u 2 XU 2 YU and the stability condition~~

~~a .u;v/ c kuk Ä sup U for all u 2 X : S XU v U 0¤v2YU k kYU~~

~~As a consequence, we conclude unique solvability of the variational problem (26.16) 2 to obtain u 2 XU D L .˝/.~~

~~26.4 Single Layer Potential~~

We now consider the single layer potential (26.2), u.x/ D .eVw/.x/, x 2 3n . inte When deﬁning g D 0 Vw, we observe that u is a solution of the Dirichlet boundary value problem (26.1) being also the unique solution of the strong Galerkin–Petrov 0 formulation (26.6). To ensure u 2 H.˝/, we chose 2 ŒH.˝/ and we consider Z Z

e 0 hVw; iH.˝/ŒH.˝/ D .x/ U .x; y/w.y/ dsy dx ˝ Z Z D w.y/ U .x; y/ .x/ dx dsy Dh'j ; wi ˝ where the duality pairing has to be speciﬁed. Using the Newton potential Z '.y/ D .N0 /.y/ D U .x; y/ .x/dx for y 2 ˝ ˝ and the Dirichlet datum .y/ D '.y/ D .N0 /.y/ for y 2 , we note that ' 2 XU is the solution of the Dirichlet boundary value problem

~~y'.y/ D .y/ for y 2 ˝; '.y/ D .y/ for y 2 :~~

~~' Œ .˝/0 0 Œ .˝/ 1.˝/0 In fact, 2 XU solves, due to 2 H YU D H \ H0 ,the ultra-weak variational formulation Z Z Z @ '.x/v.x/ dx D .x/ v.x/ dsx .x/v.x/dx for all v 2 YU: ˝ @nx ˝~~

~~This variational formulation implies h i h i 0 0 int int 1 D 'j 2 1 YU D 1 ŒH0 .˝/ \ H.˝/ 300 O. Steinbach from which we further conclude int 1 int w 2 1 ŒH0 .˝/ \ H.˝/ D 1 H.˝/ as well as~~

~~e int V W 1 H.˝/ ! H.˝/ ; and ﬁnally,~~

~~int int V W 1 H.˝/ ! 0 H.˝/ (26.17)~~

~~.˝/ s.˝/ 3 < < follows. In particular, when H Â H is satisﬁed for 2 s si,wehave~~

~~3 1 V W Hs 2 . / ! Hs 2 . / :~~

~~Since we can do the same considerations subject to the exterior problem, we ﬁnally conclude~~

~~s1 s V W H . / ! H . / for all s 2 .1; minfsi 1=2; se 1=2g/: (26.18)~~

Note that in the case of a polygonal bounded domain ˝ 2 this result was already given in [CoSt85]. Due to the mapping properties (26.18) and when assuming g 2 Hs. / for some s 2 .1; minfsi; seg/ we may consider the Galerkin–Petrov variational formulation to ﬁnd w 2 Hs1. / such that

~~s hVw;vi Dhg;vi for all v 2 H . /: (26.19)~~

To prove unique solvability of the Galerkin–Petrov formulation (26.19) we need to establish an appropriate stability condition. s1 Theorem 1 Let w 2 H . / be given for some s 2 .1; minfsi; seg/. Then there holds the stability condition

hVw;vi cV kwkHs1. / Ä sup (26.20) v 0¤v2Hs. / k kH s. / with a positive constant cV >0independent of w. Proof In fact, we follow the standard approach to prove the ellipticity estimate (26.5), see, e.g., [HsWe77, St08]. Since u D eVw is harmonic in ˝, Green’s ﬁrst formula implies Z Z @ 2 u.x/u.x/ dsx D jru.x/j dx : (26.21) @nx ˝ 26 Stability of the Laplace Single Layer Boundary Integral Operator 301

With the jump relations for the interior Dirichlet and Neumann trace operators for the single layer potential we have 1 intu.x/ .Vw/.x/; intu.x/ w.x/ .K0w/.x/ x ; 0 D 1 D 2 C for 2 where in addition to the single layer boundary integral operator V we used the adjoint double layer boundary integral operator, Z @ 0 .K w/.x/ D U .x; y/w.y/dsy; x 2 : @nx

~~Hence, (26.21)gives 1 0 2 h. I C K /w; Vwi Dkruk 2 : 2 L .˝/~~

~~When doing the same considerations subject to the exterior Dirichlet boundary value problem, this gives 1 0 2 h. I K /w; Vwi Dkruk 2 : 2 L .˝c/~~

~~Note that in 3 the single layer potential u D eVw satisﬁes the radiation condition (26.14). Now we conclude, by using (26.12) and (26.15),~~

~~; 2 2 int 2 ext 2 : hw Vwi DkrukL2.˝/ CkrukL2.˝c/ cik 1 ukHs1. / C cek 1 ukHs1. /~~

~~On the other hand, we have Á 2 int ext 2 2 int 2 ext 2 kwkHs1. / Dk 1 u 1 ukHs1. / Ä k 1 ukHs1. / Ck 1 ukHs1. / and therefore we obtain~~

2 ; 1 s hw Vwi cS kwkHs1. / cV kwkHs . /kwkH . / due to Hs1. / Hs. / for s >1, which ﬁnally gives (26.20). ut From the stability condition (26.20) we can conclude unique solvability of the Galerkin–Petrov formulation (26.19). By using

~~s1 s hVw;vi Dhw; Vvi for w 2 H . /; v 2 H . / we can deﬁne the single layer boundary integral operator V W Hs. / ! H1s. /, which satisﬁes the following stability condition. 302 O. Steinbach~~

~~s Lemma 3 Let v 2 H be given for some s 2 .1; minfsi; seg/. Then there holds the stability condition~~

hVv;wi cV kvkHs. / Ä sup (26.22) 0¤w2Hs1. / kwkHs1. / with the positive constant cV >0as used in (26.20). Proof For g 2 Hs. / we ﬁnd, by solving (26.19), w 2 Hs1. /, and the stability condition (26.20)gives

~~;v ;v hVw i hg i : cV kwkHs1. / Ä sup D sup ÄkgkHs. / v v 0¤v2Hs. / k kH s. / 0¤v2Hs. / k kH s. /~~

~~Using duality we then conclude the stability estimate~~

hv;gi 1 hv;Vwi kvkHs. / D sup Ä sup 0¤g2Hs. / kgkHs. / cV 0¤w2Hs1. / kwkHs1. / as claimed. ut Using the indirect single layer potential u.x/ D .eVv/.x/, x 2 ˝ we can describe the solution of the Dirichlet boundary value problem (26.1) with a given Dirichlet datum g 2 H1s. /, i.e. v 2 Hs. / is the unique solution of the Galerkin–Petrov formulation

~~s1 hVv;wi Dhg; wi for all w 2 H . /: (26.23)~~

Due to s >1it is possible to consider g 2 L2. / H1s. / within the variational formulation (26.23) which can be seen as the boundary integral equation counter part of the ultra-weak ﬁnite element formulation [ApEtAl16]. Remark 1 In the two-dimensional case ˝ 2 we need to have w 2 Hs1. / with the constraint hw;1i D 0 to satisfy the radiation condition (26.14) for the single layer potential u.x/ D .eVw/.x/ as jxj!1. To ensure solvability of the Galerkin– Petrov formulation (26.19), g has to satisfy the solvability condition hg; weqi D 1 0 with the natural density weq D V 1. So solvability of the Dirichlet boundary value problem (26.1) for general g can always be guaranteed when considering an hg;w i appropriate additive splitting of g.x/ D C eg.x/, D eq , where we have g g h1;weqi to assume diam ˝<1to ensure h1; weqi >0. All other results then remain true when considering appropriate factor spaces. 26 Stability of the Laplace Single Layer Boundary Integral Operator 303

~~References~~

[ApEtAl16] Apel, T., Nicaise, S., Pfefferer, J.: Discretization of the Poisson equation with non- smooth boundary data and emphasis on non-convex domains. Numer. Methods PDE 32, 1433–1454 (2016) [Co88] Costabel, M.: Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal. 19, 613–626 (1988) [Co90] Costabel, M.: Boundary integral operators for the heat equation. Integr. Equ. Oper. Theory 13, 498–552 (1990) [CoSt85] Costabel, M., Stephan, E.P.: Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximations. In: Mathematical Models and Methods in Mechanics. Banach Centre Publ., vol. 15, pp. 175–251. PWN, Warschau (1985) [Gr92] Grisvard, P.: Singularities in Boundary Value Problems. Research Notes in Applied Mathematics, vol. 22. Springer, New York (1992) [HsWe77] Hsiao, G.C., Wendland, W.L.: A ﬁnite element method for some integral equations of the ﬁrst kind. J. Math. Anal. Appl. 19, 449–481 (1977) [St08] Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements. Springer, New York (2008) [Ve84] Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984) Chapter 27 Spectral Lanczos’ Tau Method for Systems of Nonlinear Integro-Differential Equations

~~P.B. Vasconcelos, J. Matos, and M.S. Trindade~~

~~27.1 Introduction~~

The tau method is a spectral method, originally developed by Lanczos in the 1930s [La38] that delivers polynomial approximations to the solution of differential problems. The method tackles both initial and boundary value problems with ease. It is a spectral method thus ensuring excellent error properties, whenever the solution is smooth. Initially developed for linear differential problems with polynomial coefficients, it has been used to solve broader mathematical formulations: functional coefficients, nonlinear differential and integro-differential equations. Several studies applying the tau method have been performed to approximate the solution of differential linear and nonlinear equations [CrRu83, LiPa99], partial differential equations [OrDi87, MaEtAl04], and integro-differential equations [MaEtAl05, AbTa09], among others. Nevertheless, in all these works the tau method is tuned for the approximation of specific problems and not offered as a general purpose numerical tool.

P.B. Vasconcelos () Center of Mathematics, University of Porto, Porto, Portugal Economics Faculty, University of Porto, Porto, Portugal e-mail: [email protected] J. Matos Center of Mathematics, University of Porto, Porto, Portugal Polytechnic School of Engineering Porto, Porto, Portugal e-mail: [email protected] M.S. Trindade Center of Mathematics, University of Porto, Porto, Portugal e-mail: marcelo.trindade.fc.up.pt

~~© Springer International Publishing AG 2017 305 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_27 306 P.B. Vasconcelos et al.~~

A barrier to use the method as a general purpose technique has been the lack of automatic mechanisms to translate the integro-differential problem by an algebraic one. Furthermore and most importantly, problems often require high- order polynomial approximations, which brings numerical instability issues. The tau method inherits numerical instabilities from the large condition number associated with large matrices representing algebraically the actions of the integral, differential or integro-differential, operator on the coefficients of the series solution. In this work numerical instabilities related with high-order polynomial approximations, in the tau method, are tackled allowing for the deployment of a general framework to solve integro-differential problems. We aim at contributing to broad- cast the tau method for the scientific community and industry as it provides polynomials solutions with good error properties. The Tau Toolbox [TrEtAl16] is a MATLAB tool to solve integro-differential problems by the tau method. It aggregates all contributions available, enhances the use of the method by developing more stable algorithms, and offers efficient implementations.

~~27.2 Preliminaries~~

We begin by introducing the notation for the algebraic formulation of the tau method. Assume throughout that P D ŒP0; P1;::: Â is an orthogonal basis for the 2 polynomials space of any non-negative integer degree,PX D Œ1; x; x ;:::2 the . / P power basis for . Furthermore, consider that y x D i0 aiPi D aPis a formal Œ ; ;:::T . / i series with coefﬁcients a D a0 a1 . For the power basis, y x D i0 aix D XaX. Lemma 1 illustrates matrices M, N, and O that set, respectively, polynomial multiplication, differentiation, and integration into algebraic operations. 1 Lemma 1 Let V be the triangularR matrix such that P D XV and a D V aX. Then P d P P xy D Ma, dx y D Na, and ydx D Oa where

~~1 1 1 M D V MXV; N D V NXV and O D V OXV; (27.1) 2 3 2 3 2 3 0 01 0 6 7 6 7 6 7 610 7 6 02 7 610 7 MX D 6 10 7 ; NX D 6 03 7 and OX D 6 1 0 7 : 4 5 4 5 4 2 5 : : : : : : :: :: :: :: :: ::~~

Proof See [OrSa81]forM and N. It is then easy to extend to O (see [HoSh03]). The next proposition shows how to translate a linear ordinary differential and integral operators, with polynomial coefﬁcients, into an algebraic representation. 27 Spectral Tau Method for Nonlinear Integro-Differential Equations 307

Proposition 1 The th order, 2 , ordinary linear differential operator Dy D P k p d y and the th order, 2 , ordinary linear integral operator Sy D PkD0 k dxRk ` P `D0 p` ydx acting on , are casted on by, respectively, X k Dy D PDa; D D pk.M/N (27.2) kD0 and X ` Sy D PSa; S D p`.M/O ; (27.3) `D0 P . / nr i ;` with pr M D iD0 pr;iM ,rD k and nr 2 0. Proof Note that Z d XMXaX D xy; XNXaX D y and XOXaX D ydx dx for MX, NX, and OX as in Lemma 1. Then, X X 1 P (i) xy D MXaX D VMV aX D MaI P d 1 dky X X X X X P D P (ii) dx y D N a D VNV a D Na and thus y D kD0 pk dxk D DaI R P R ` X X 1 P S (iii) ydx D OXaX D VOV aX D Oa and thus y D `D0 p` ydx D PSa: Let K.x; t/ be a two-variable polynomial, or aˇ two-variable polynomial approx- ˇ P Œ 0. /; 1. /;::: imation of a two-variable function. Then, for xDt D P t P t and K 2 nxnt ,

Xnx Xnt ˇ T ˇ . ; / ; . / . / P P : K x t D ki jPi x Pj t D K xDt (27.4) iD0 jD0 R R x x Lemma 2 For the integral operator PK.x; tP/y.t/dt, where stands for the . ; / nx nt . / . / . / P calculationR of the integral at x, K x t D iPD0 jPD0 kijPi x Pj t and y x D a, x . ; / . / P ; nx nt . / . /: one has K x t y t dt D Sa where S D iD0 jD0 kijPi M OPj M Proof Using Lemma 1 one can deduce that Z x xitjy.t/dt D X.VMiV1/.VOV1/.VMjV1/Va

D PMiOMja; and therefore Z x Pi.x/Pj.t/y.t/dt D PPi.M/OPj.M/a: 308 P.B. Vasconcelos et al. R ˇ x ˇ Lemma 3 P .x/P .t/y.t/dt D P.P .M/ e 1P /OP .M/a; where e 1 is x0 i j i iC xDx0 j iC the .i C 1/th column of the identity matrix. Proof From Lemma 2 Z x ˇ P .x/P .t/y.t/dt D PP .M/OP .M/a Pˇ P .M/OP .M/a i j i j xDx0 i j x0 ˇ ˇ D P.P .M/ e 1P /OP .M/a: i iC xDx0 j ˇ ˇ ˇ i ˇ Note that it is easy to understand that X M D e 1X . xDx0 X iC xDx0 R ˇ ˇ b ˇ ˇ . / . / . / P 1.P P / . / : Lemma 4 a Pi x Pj t y t dt D eiC xDb xDa OPj M a Proof Immediate since it is a particular case of Lemma 3. R Proposition 2 The linear Volterra integral operator S y D x K.x; t/y.t/dt and R V x0 S b . ; / . / the FredholmP integralP operator Fy D a K x t y t dt, with degenerate kernel . ; / nx nt . / . / K x t iD0 jD0 kijPi x Pj t , smooth and continuous, acting on y have, respectively, the following algebraic representation

~~Xnx Xnt ˇ Á ˇ S y D k P .M/ e 1P OP .M/a (27.5) V ij i iC xDx0 j iD0 jD0 and~~

Xnx Xnt ˇ ˇ Á ˇ ˇ SFy D kijeiC1 P Pˇ OPj.M/a (27.6) xDb xDa iD0 jD0 for ei the ith column of the identity matrix. Proof Equation (27.5) can be readily obtained from Lemma 3 (see, e.g., [SaEtAl13]) and equation (27.6) from Lemma 4 (see e.g. [HoSh03]).

~~27.3 The Tau Method for Integro-Differential Problems~~

An approximate polynomial solution yn for the linear integro-differential problem ( Dy C Sy C S y C S y D f V F ; (27.7) ci.y/ D si; i D 1;:::; is obtained in the tau sense by solving a perturbed system ( Dy C Sy C S y C S y D f C n n V n F n n ; (27.8) ci.yn/ D si; i D 1;:::; 27 Spectral Tau Method for Nonlinear Integro-Differential Equations 309 where f is a th degree polynomial (or a polynomial approximation of a function), n is the residual and ci.y/ D si, i D 1;:::; the initial and/or boundary conditions. Problem (27.7) has a matrix representation given by ( Ca D s ; (27.9) .D C S C SV C SF/a D f

T where C D Œcij 1; cij D ci.Pj1/; i D 1;:::; ; j D 1;2;:::, a D Œa0; a1;::: T the coefficients of y in P, s D Œs1;:::;s , D, S, SV , and SF as defined in, T respectively, (27.2)–(27.6), and f D Œf0;:::;f;0;0;::: the right-hand side of the system. Choosing an integer n C ,an.n 1/th degree polynomial approximate solution yn D Pnan is obtained by truncating system (27.9) to its first n columns. Moreover, restricting this system to its first n C C h equations, a linear system of dimension n n is obtained, which is equivalent to introduce a polynomial residual

~~n D .Dy Dyn/ C .Sy Syn/ C .SV y SV yn/ C .SFy SFyn/: (27.10)~~

~~27.4 Nonlinear Approach for Integro-Differential Problems~~

~~Nonlinear differential problems are tackled with linear approximations and solving a set of linear problems. Let G be the nonlinear operator acting on an appropriate space of smooth functions~~

G.y. /;:::;y.1/; y.0/; y.1/;:::;y. // D 0; (27.11) Z d`y where y.`/ D for ` 2 C, y.`/ D . ydt/` for ` 2 and includes Volterra dx` and Fredholm terms. If G is ¼1 in a neighborhood ˝ of ! D .y. /;:::;y.1/; y.0/; y.1/;:::;y. // and if !0 2 ˝ is an approximation of !, then a linear operator T can be deﬁned, represented by the order one Taylor polynomial centered at !0

~~X @ G .i/ .i/ T.!/ D G.!0/ C j! .y y / @y.i/ 0 0 iD~~

~~As in the Newton method for algebraic equations, we can replace G by T in (27.11) and solve the approximated equation~~

~~X @ X @ G .i/ G .i/ j! y DG.!0/ C j! y : (27.12) @y.i/ 0 @y.i/ 0 0 iD iD 310 P.B. Vasconcelos et al.~~

Applying the Tau method to the linear differential equation (27.12) and taking . / . / !1 D .y1 ;:::;y1 / as the solution, if !1 2 ˝ we can repeat the process, obtaining an iterative procedure, solving for !k the linear differential equation X X @G .i/ @G .i/ j! y DG.! 1/ C j! y ; k D 1;2;:::: @y.i/ k1 k k @y.i/ k1 k1 iD iD

~~27.5 Contributions to Stability~~

In this section we summarize some of the mathematical techniques developed for the tau method to provide stable algorithms for the Tau Toolbox library. Let P D ŒP0.x/; P1.x/;::: be an orthogonal basis satisfying xPj D ˛jPjC1 C ˇjPj C jPj1; j 0; P0 D 1; P1 D 0. Orthogonal Evaluation: If P are the corresponding orthogonalP polynomials Œ ; . / n shifted to a b and x is a vector, then the evaluation of yn x D iD0 aiPi is directly computed in P by the recursive relation 8 ˆ T <ˆP0 D Œ1;:::;1 c1xC.c2ˇ0/P0 P1 D ˛0 ˆ : . 1 C 2ˇ 1 /ˇ 1 P D c x c i P0 Pi1 i Pi2 ; i D 2;3;:::;n i ˛i1

~~2 aCb where ˇ is the element-wise product of two vectors, c1 D ba and c2 D ab . The Tau Toolbox proposes an orthoval function, instead of the polyval MATLAB one, to implement this functionality.~~

Change of Basis by Recurrence: Let V satisfy aX D Va. The coefﬁcients of W D V1 can be computed without inverting V by the recurrence relation ( w1 D e1 ; (27.13) wjC1 D Mwj; j D 1;2;::: where M is such that Px D PM, wj is the jth column of M, and e1 the ﬁrst column of the identity matrix. Avoiding Similarity Transformations: Matrix inversion presented at all similarity transformations must be avoided to ensure numerically stable computations. Recur- rence relations to compute the elements of matrices M, N, and O directly on P can be computed, respectively, by 27 Spectral Tau Method for Nonlinear Integro-Differential Equations 311 ( n jC1;j D ˛j1;j;j D ˇj1;j;jC1 D j1 ; ; M D i j i;jD1 D i;j D 0; ji jj >1 8 < ˛i1j;i1C.ˇiˇj/j;iC iC1j;iC1 jj1;i i;jC1 D n ˛j N D i;j D ; and i;jD1 : ˛j1j;j1C1 j;jC1 D ˛j ( ˛j 1; D n jC j jC1 : O D i;j ; 1 D P 1 i jD ai jC ; 1;:::;1;0 iC1;j DiC1 kDiC2 i;k k;j i D j

Linearization Coefficients: Product of polynomials p and q in P occurs in nonlinear problems. Usually both polynomials are translated first from P to X, then the convolution is applied and finally the product is translated back, pq D 1 V conv.Vp; Vq/. Alternatively, ensuring robustness, thisP product can be directly P iCj computed on using the linearization coefficients: PiPj D kD0 li;j;kPk, where li;j;k are computed by recurrence relations [Le96]. For p D Pa and q D Pb,ofdegreen, then pq D Pc where 8 ˆ Xk Xnj 1 ˆ ıi;j ˆck D . / .aibj C ajbi/li;iCj;k; k D 0;:::;n <ˆ 2 jD0 kC1j iDb 2 c : ˆ Xnk Xnj 1 ˆ ı ; ˆc D . / i j .a b C a b /l ; ; ; k D 1;:::;n :ˆ nCk 2 i j j i i iCj nCk jD0 nkj iDb 2 c

Computing with M: An efﬁcient way to compute the powers of M is k Œ.k/; .k/ .k1/˛ .k1/ˇ .k1/ : M D i;j i;j D i1;j i1 C i;j i C iC1;j iC1 P . / n . / Moreover, the evaluation of yn M D iD0 an;iPi M can be performed with

~~.M ˇjI/Pj.M/ jPj1.M/ PjC1.M/ D ; j 0; ˛j~~

~~P0.M/ D I and P1.M/ D ¿.~~

~~27.6 Numerical Results~~

Example 1 Consider the integro-differential Fredholm nonlinear equation of the second kind [DeSa12], with exact solution y.x/ D exp.x/, Z d 1 y.x/ C y.x/ y.t/2dt D 0:5.e2 1/; y.0/ D 1: (27.14) dx 0 312 P.B. Vasconcelos et al.

2 d 2 d Introducing new variables y1 D y and y2 D y1, we get dx y2 D y1 dx y1. Linearizing d .k/ d d .k/ .k/ d .k/ it results y1 dx y1 y1 dx y1 C dx y1 y1 y1 dx y1 and, therefore, problem (27.14) can be casted as

~~8 Z 1 d . 1/ . 1/ . 1/ ˆ kC kC kC 0:5. 2 1/ ˆ y1 C y1 y2 dt D e~~

Tau Toolbox fred where SFT by (27.6), makesh use of the function.i The T 1; 1; 0:5. 2 1/; 0; :::; 0; 2 .k/ d .k/ : independent vector is bT D e y1 dx y1 The error presented in [DeSa12]isjjejj1 D maxj jyn.xj/ y.xj/j, and therefore the same measure is applied to compare the results, see Table 27.1. For all polynomial degree approximations the approximate solution provided by Tau Toolbox is clearly better than the one given by [DeSa12]. The error for n D 129 with [DeSa12] was reached with Tau Toolbox with a polynomial degree smaller than 9 and for degree 17 the Tau Toolbox was able to ﬁnd the solution with machine precision order. Noteworthy is that for increasing polynomial degree the algorithm is stable, not showing any perturbations for higher degrees. The CPU time should not be compared between both approaches since results are reported from two distinct machines. It is however relevant to understand that the effort to solve the problem with Tau Toolbox is higher: if we take n D 5 as reference time then for n D 128 Tau Toolbox required 78:3 the reference computational cost whereas [DeSa12] necessitates 5:12. Robustness and stability

Table 27.1 Comparison [DeSa12] Tau Toolbox between the results at n jjejj1 CPU time jjejj1 CPU time [DeSa12]andwithTau Toolbox. 5 9.63e-04 0.42 1.58e-04 0.03 9 1.28e-04 0.58 1.28e-09 0.03 17 2.87e-05 0.73 7.77e-16 0.04 33 5.61e-06 0.07 4.44e-16 0.07 65 2.39e-06 1.54 4.44e-16 0.37 129 1.28e-06 2.15 4.44e-16 2.35 27 Spectral Tau Method for Nonlinear Integro-Differential Equations 313 comes at a price: more elaborate mathematics must be performed to reach such quality results. Nevertheless, one must point out that the CPU time to compute an accurate approximation was still low, only 2:35 seconds. Example 2 Consider now the system of integro-differential equations with nonlinear Volterra term [AbTa09], with exact solution y1.x/ D sinh.x/ and y2.x/ D cosh.x/, 8 Â Ã Z 2 x ˆ d 1 d ˆ y1.x/ C y2.x/ .x t/y2.t/ C y2.t/y1.t/dt D 1

As for the previous example, linearization is done ﬁrst and the Volterra integral term SV , following (27.5), is tackled using the Tau Toolbox volt function. Figure 27.1 shows the error after 5 iterations along the interval Œ0; 1.For n D 20, Tau Toolbox was able to provide an approximate solution with machine precision all over the interval. For comparison purposes, results for the same problem in [AbTa09] with n D 10 are plotted together with Tau Toolbox library. The former, at the right part of the interval, can only reach single-precision accuracy, in contrast with the latter, which delivers double precision accuracy. For n 25, Tau Toolbox reaches machine precision.

~~10-5 10-8~~

~~10-10 10-10~~

~~10-12~~

~~10-15 10-14~~

~~10-20 10-16 0 0.2 0.4 0.6 0.81 0 0.2 0.4 0.6 0.81~~

~~Fig. 27.1 Comparison between [AbTa09]andTau Toolbox for 10 (and 20) degree polynomial. 314 P.B. Vasconcelos et al.~~

~~27.7 Conclusions~~

In this work we proposed the Lanczos’ tau method for nonlinear integro-differential systems of equations. Contributions to improve the stability of the numerical implementation are presented. Numerical experiments illustrate the accuracy and efﬁciency of the new proposal, when compared with the results in the literature. All these contributions are included at Tau Toolbox –aMATLAB library for the solution of integro-differential problems.

Acknowledgment This work was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER) under the partnership agreement PT2020. The third author was supported by CAPES, Coordination of Superior Level Staff Improvement - Brazil.

~~References~~

[AbTa09] Abbasbandy, S., Taati, A.: Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation. J. Comput. Appl. Math. 231(1), 106–113 (2009) [CrRu83] Crisci, M.R., Russo, E.: An extension of Ortiz recursive formulation of the Tau method to certain linear systems of ordinary differential equations. Math. Comput. 38(9), 27–42 (1983) [DeSa12] Dehghan, M., Salehi, R.: The numerical solution of the non-linear integro-differential equations based on the meshless method. J. Comput. Appl. Math. 236(9), 2367–2377 (2012) [HoSh03] Hosseini, S.M., Shahmorad, S.: Numerical solution of a class of integro-differential equations by the Tau method with an error estimation. Appl. Math. Comput. 136, 559–570 (2003) [La38] Lanczos, C.: Trigonometric interpolation of empirical and analitical functions. J. Maths. Phys. 17, 123–199 (1938) [Le96] Lewanowicz, S.: Second-order recurrence relation for the linearization coefﬁcients of the classical orthogonal polynomials. J. Comput. Appl. Math. 69(1), 159–170 (1996) [LiPa99] Liu, K.M., Pan, C.K.: The automatic solution to systems of ordinary differential equations by the Tau method. Comput. Math. Appl. 38(9), 197–210 (1999) [MaEtAl04] Matos, J., Rodrigues, M.J., Vasconcelos, P.B.: New implementation of the Tau method for PDEs. J. Comput. Appl. Math. 164–165, 555–567 (2004) [OrSa81] Ortiz, E.L., Samara, H.: An operational approach to the tau method for the numerical solution of nonlinear differential equations. Computing 27, 15–25 (1981) [OrDi87] Ortiz, E.L., Dinh, A.P.N.: Linear recursive schemes associated with some nonlinear partial differential equations in one dimension and the Tau method. SIAM J. Math. Anal. 18(2), 97–119 (1987) [MaEtAl05] Pour-Mahmoud, J., Rahimi-Ardabili, M.Y., Shahmorad, S.: Numerical solution of the system of Fredholm integro-differential equations by the Tau method. Appl. Math. Comput. 168(1), 465–478 (2005) [SaEtAl13] Saeedi, L., Tari, A., Masuleh, S.H.M.: Numerical solution of some nonlinear Volterra integral equations of the ﬁrst kind. Appl. Appl. Math. 8(1), 214–226 (2013) [TrEtAl16] Trindade, M., Matos, J., Vasconcelos, P.B.: Towards a Lanczos’ -method toolkit for differential problems. Math. Comput. Sci. 10(3), 313–329 (2016) Chapter 28 Discreteness, Periodicity, Holomorphy, and Factorization

~~V.B. Vasilyev~~

~~28.1 Introduction~~

The main topic of the paper is to establish some relations between the solvability of a special kind of discrete equations in certain canonical domains and holomorphy properties of their Fourier analogues. We start from the theory of pseudo-differential operators and equations [Ta81, Tr80, Sh01], and corresponding boundary value problems [Es81] and we shall try to construct a discrete analogue of this theory with forthcoming limit passage from discrete case to a continuous one. Historically, the theory of pseudo-differential operators started from a special kind of integral operators, namely Calderon–Zygmund operators of the following type Z .Ku/.x/ D p:v: K.x; x y/u.y/dy; (28.1)

m where the kernel K.x; y/ has certain speciﬁcal properties [MiPr86]. If we enlarge the class of such kernels and in particular we permit that the kernel K.x; y/ can be a distribution on a second variable y, then the above formula will include differential operators with variable coefﬁcients

~~Xn @ku .Du/.x/ D ak.x/ .x/; @ k1 @ km jkjD0 x1 xm where k is a multi-index, jkjDk1 Ckm.~~

~~V.B. Vasilyev () National Research Belgorod State University, Belgorod, Russia e-mail: [email protected]~~

~~© Springer International Publishing AG 2017 315 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_28 316 V.B. Vasilyev~~

~~Indeed for this case we use the kernel~~

~~Xn k1 km K.x; y/ D ak.x/ı .y1/ ı .ym/: jkjD0~~

A theory for similar operators consists in a description of functional spaces in which these operators are bounded, possible additional conditions (maybe boundary conditions) which permit to state the well-posedness of boundary value problem in a corresponding functional space and so on. Here, we would like to discuss these problems for a discrete situations and to describe some of our ﬁrst results in this direction.

~~28.2 Discreteness~~

m We consider functions ud of a discrete variable xQ 2 h , where h >0is a small parameter, and operators deﬁned on such functions of the following type X m .Adud/.xQ/ D aud.xQ/ C Ad.xQ Qy/ud.yQ/h ; xQ 2 Dd; (28.2)

~~yQ2Dd taking partial sums of the series (28.2) over cubes~~

m QN DfQx 2 h W max jxkjÄNg; 1ÄkÄm where we use following notations. m m Let D be a domain, Dd Á D\h be a discrete set, Ad be a given function m of a discrete variable deﬁned on h , and a 2 ¼. We say that the function Ad.xQ/ is the kernel of the discrete operator Ad. This kernel may be summable, i.e. it can be generated by integrable function Z jA.x/jdx < C1;

m but for this case we deal with ordinary convolution. The author has considered the more interesting and complicated case when the generating function A.x/ is a Calderon–Zygmund kernel [VaEtAl15-2, VaEtAl15-3]. These Calderon–Zygmund operators play an important role as the simplest model of a pseudo-differential operator [MiPr86, Es81]. Taking into account our forthcoming considerations of discrete pseudo-differential operators we shall restrict to this simplest case. 28 Discreteness, Periodicity, Holomorphy, and Factorization 317

We assume that our generating function A.x/ is a Calderon–Zygmund kernel, i.e. it is homogeneous of order m and has vanishing mean value on unit sphere Sm1 m, also it is continuously differentiable out of the origin and by deﬁnition A.0/ D 0. The ﬁrst question which arises in this situation is the following. Is there a certain dependence on a parameter h for a norm of the operator Ad ? Fortunately the answer is negative (see also [VaEtAl15-3] for the whole space m). Theorem 1 Let D be a bounded domain in m with a Lipschitz boundary @D. Then the norm of the operator Ad W L2.Dd/ ! L2.Dd/ doesn’t depend on h. This property leaves us hope to describe the spectra of the operator Ad using methods developed for this purpose in a continuous case.

~~28.3 Periodicity~~

Roughly speaking the Fourier image of the lattice hm is a periodic structure with 1 m h basic cube of periods „ , where „D 2 . More precisely if we introduce a discrete Fourier transform by the formula X ixQ m eud./ Á .Fdud/./ D e ud.xQ/h (28.3) xQ2hm taking partial sums of the series (28.3) over cubes QN , we can use this construction to give a deﬁnition of a discrete pseudo-differential operator by the formula Z ixQe .Adud/.xQ/ D e Ad./eud./d;

„m e where the function Ad./ is called the symbol of the operator Ad. Let us note that this discrete Fourier transform preserves all basic properties of standard Fourier transform. Only one principal distinction is periodicity of Fourier images. e Deﬁnition 1 The symbol Ad./ is called an elliptic symbol (and operator Ad is e called an elliptic one) if ess inf jAd./j >0. 2„m m m Proposition 1 The operator Ad W L2.h / ! L2.h / is invertible iff it is an elliptic operator. Many interesting properties of the operator Ad related to a comparison between continuous and discrete cases can be found in [VaEtAl15-2]. 318 V.B. Vasilyev

~~28.4 Holomorphy~~

~~The property of holomorphy arises if we try to obtain a Fourier image for a so-called paired equation~~

.AdPC C BdP/Ud D Vd; (28.4) where P˙ are projectors on some canonical domains (see below), Ad; Bd are discrete operators similar to (28.2). m m If, for example, P˙ are projectors on discrete half-spaces h ˙ DfQx 2 h W xQ D .xQ1; ; xQm/; ˙Qxm >0g, and we want to use standard properties of the Fourier transform related to a convolution then, in order to ﬁnd a Fourier image of the . / . / m product C xQ U xQ where C is an indicator of the h C, we need to go out in a complex domain [VaEtAl15-1, VaEtAl15-3]. 0 We introduce for ﬁxed D .1; ;m1/

~~1 ˘˙ Dfm ˙ i 2 ¼ W m 2 h Œ ; ; > 0g:~~

m Theorem 2 Let H˙ be subspaces of the space L2.„ / consisting of functions which admit holomorphic extensions into upper and lower complex half-strips ˘˙ 0 on a last variable m under almost all ﬁxed D .1; ;m1/. Then we have the following decomposition

~~m L2.„ / D HC ˚ H:~~

~~Indeed the decomposition is given by the following operators~~

~~Zh1 1 . / per 0 h t m .H 0 uQ /. / D p:v: uQ . ; t/ cot dt; d m 2 i d 2 h1~~

~~per per per per P0 D 1=2.I C H0 /; Q0 D 1=2.I H0 /; so that~~

~~per per FdPCud D P0 eud; FdPud D Q0 eud:~~

~~28.5 Factorization~~

The concept of factorization is needed if we consider an original equation in a non- whole lattice, i.e. D ¤ m. We extract from m some so-called canonical domains. The fact that to obtain Fredholm conditions for an elliptic operator (or equation) on a manifold, and in particular in a domain of m-dimensional space, we need to 28 Discreteness, Periodicity, Holomorphy, and Factorization 319 obtain an invertibility conditions for a local representative of original operator, is called a local principle [MiPr86, Va00]. Roughly speaking such local representatives are simple model operators in canonical domains. If for example we are interested in studying a Calderon–Zygmund operator on a manifold with a boundary of the type (28.1), we need to describe invertibility conditions for the following model operators in the following canonical domains:

~~• for inner points x0 of a manifold Z~~

~~u.x/ 7! p:v: K.x0; x y/u.y/dy;~~

~~• for boundary points x0 on smooth parts of a boundary Z~~

~~u.x/ 7! p:v: K.x0; x y/u.y/dy;~~

~~m C~~

m m . ; ; /; >0 where C Dfx 2 W x D x1 xm xm g, a • for boundary points x0 for which their neighborhood is diffeomorphic to CC D m 0 0 0 fx 2 W x D .x ; xm/; x D .x1; ; xm1/; xm > ajx j; a >0g Z

~~u.x/ 7! p:v: K.x0; x y/u.y/dy:~~

~~a CC~~

It is natural to expect similar properties for general discrete operators. That’s why we consider here the simplest model operators in cones. So we have the following m; m ; a canonical domains: C CC. It is essential that all these domains are cones but the first two include a whole straight line. The case D D m is very simple (from modern point of view; there was a lot of mathematicians whose papers have helped us to clarify this situation). If a symbol e Ad./ of the operator Ad from (28.2) is elliptic, then such operator Ad is invertible m at least in the space L2.h /. We apply the discrete Fourier transform (28.3) and obtain immediately that the operator Ad is unitary equivalent to a multiplication operator on its symbol. We proceed to describe the half-space case. First we recall the following definition. e Definition 2 Factorization of an elliptic symbol Ad./ is called its representation in the form

~~e ./ eC./ e./; Ad D Ad Ad~~

~~e˙./ where the factors Ad admit a bounded holomorphic continuation into upper and 0 m1 lower complex half-strips ˘˙ for almost all D .1; ;m1/ 2 . 320 V.B. Vasilyev~~

~~Now we come back to the equation (28.4), we apply the discrete Fourier transform Fd and we obtain the following equation~~

~~eA .0; / C eB .0; / d m d m UQ ./ 2 d C~~

Zh1 eA .0; / eB .0; / h. / d m d m p:v: UQ .0;/cot m d D VQ ./: (28.5) 4 i d 2 d h1 e e where Ad./; Bd./ are symbols of discrete operators Ad; Bd. Of course, equation (28.5) is related to the corresponding Riemann boundary value problem [Ga81, Mu76, VaEtAl13, VaEtAl15-1, VaEtAl15-3], so the following result is valid. Theorem 3 For m 3 the equation (28.5) is uniquely solvable in the space m L2.h / iff operators Ad; Bd are elliptic and

~~e . ; /e1. ; / 0: Ind Ad m Bd m D~~

~~The key role for a proof of the theorem is played by the concept of factorization per for an elliptic symbol, and it can be constructed exactly by means of operator H0 [VaEtAl15-1, VaEtAl15-3].~~

~~28.5.1 Conical Case~~

~~This section is devoted to the last and most complicated case. Let C.xQ/ be a characteristic function of the discrete cone Dd and Sd.z/ be the following function~~

~~X ixQz Sd.z/ D C.xQ/e ; z 2 T.D/; z D C i;~~

~~xQ2Dd~~

where DDfx 2 m W x y >0;8y 2 Dg, T.D/ is a specific domain in a multidimensional complex space ¼m so that T.D/ D„m C iD. The infinite sum exists for ¤ 0 but does not exist for D 0 because it is formally the discrete Fourier transform (28.3) of the nonsummable indicator C. m If we fix a certain function ud 2 L2.h /, then we have C ı ud 2 L2.Dd/, and m therefore the discrete Fourier transform BC ı ud is defined and belongs to L2.„ /. So, according to properties of the discrete Fourier transform (28.3), we have Z

~~.Fd.C ı u//./ D lim Sd.z y/uQd.y/dy; !0C „m and the last integral exists at least in L2-sense. 28 Discreteness, Periodicity, Holomorphy, and Factorization 321~~

Thus we study a corresponding analogue of the equation (28.4) and we also need complex variables and relevant analogue of Riemann boundary value problem [BoMa48, Vl07, Va00]. m m Let A.„ / be a subspace of L2.„ / consisting of functions which admit an analytical extension into T.D/, and B.„m/ is an orthogonal complementation of m m the subspace A.„ / in L2.„ / so that

~~m m m L2.„ / D A.„ / ˚ B.„ /:~~

~~First of all we deal with a jump problem formulated in the following way: ﬁnding a pair of functions ˚ ˙;˚C 2 A.„m/; ˚ 2 B.„m/; such that~~

~~˚ C./ ˚ ./ D g./; 2„m; (28.6)~~

~~m where g./ 2 L2.„ / is given. m m Proposition 2 The operator Sd W L2.„ / ! A.„ / is a bounded projector. A m function ud 2 L2.Dd/ iff its Fourier transform uQd 2 A.„ /.~~

~~Proof According to standard properties of the discrete Fourier transform Fd we have Z~~

Fd.C.xQ/ud.xQ// D lim Sd.z /eud./d; !0 „m where C.xQ/ is the indicator of the set Dd. It implies the boundedness of the operator Bd. The second assertion follows from holomorphic properties of the kernel Sd.z/. In other words for arbitrary function v 2 A.„m/ we have Z v.z/ D Sd.z /v./d; z 2 T.D/:

~~„m~~

~~It is an analogue of the Cauchy integral formula. ut Theorem 4 The jump problem has unique solution for arbitrary right-hand side m from L2.„ /.~~

Proof Indeed there is an equivalent unique representation of the space L2.Dd/ as a direct sum of two subspaces. If we denote C.x/; .x/ the indicators of the discrete m sets Dd; h n Dd, respectively, then the following representation

~~ud.xQ/ D C.xQ/ud.xQ/ C .xQ/ud.xQ/~~

~~m is unique and holds for an arbitrary function ud 2 L2.h /. After applying the discrete Fourier transform we have~~

~~Fdud D Fd.Cud/ C Fd.ud/; 322 V.B. Vasilyev~~

m where Fd.Cud/ 2 A.„ / according to the proposition 2, and thus Fd.ud/ D m m Fdud Fd.Cud/ 2 B.„ / because Fdud 2 L2.h /. 2 2 2 Example 1 If m D and CC is the ﬁrst quadrant of , then a solution of a jump problem is given by formulas

~~Z Z 1 1 i1 t1 2 i2 t2 ˚ C./ C C . ; / D 2 lim cot cot g t1 t2 dt1dt2 .4 i/ !0 2 2~~

~~˚ ./ ˚ C./ ./; . ; / 2 : D g D 1 2 2 CC~~

The last decomposition will help us to formulate the periodic Riemann boundary value problem which is very distinct for one-dimensional case and multidimensional one. The principal non-correspondence is that the subspace B.„m/ consists of boundary values of certain analytical functions in one-dimensional case, but this set has an unknown nature for a multidimensional case. A multidimensional periodic variant of Riemann boundary value problem can be formulated as follows: ﬁnding two functions ˚ ˙./ such that ˚ C./ 2 A.„m/, ˚ ./ 2 B.„m/ and the following linear relation holds

˚ C./ D G./˚ ./ C g./; (28.7) where G./; g./ are given functions on „m. We assume here that G./ 2 C.„m/, G./ ¤ 0; 8 2„m. Deﬁnition 3 Periodic wave factorization of a function G./ is called its representation in the form

~~G./ D G¤./GD./;~~

˙1./; ˙1./ where factors G¤ GD admit a bounded analytical continuation into complex domains T.D/; T. D/, respectively. Theorem 5 If G./ admits periodic wave factorization, then multidimensional Riemann boundary value problem has a unique solution for arbitrary right-hand m side g./ 2 L2.„ /. Proof We rewrite a multidimensional Riemann boundary value problem in the form

~~1./˚ C./ ./˚ ./ 1./ ./ G¤ GD D G¤ g and obtain a jump problem (28.6). 28 Discreteness, Periodicity, Holomorphy, and Factorization 323~~

~~m m Indeed for arbitrary two functions f ; g 2 L2.h / such that supp f h n .Dd/; supp g .Dd/ according to properties of discrete Fourier transform Fd we have~~

~~. 1. //. / .. 1 / . 1 //. / Fd f ı g xQ D Fd f Fd g xQ Á X X f1.xQ Qy/g1.yQ/ D f1.xQ Qy/g1.yQ/; m yQ2h yQ2Dd~~

1 ; 1 where f1 D Fd f g1 D Fd g and according to the proposition 2 supp g1 Dd. m Further since we have supp f1 h n .Dd/ then for xQ 2 Dd; yQ 2Dd we have m xQ Qy 2 Dd so that f1.xQ Qy/ D 0 for such xQ; yQ. Thus supp .f1 g1/ h n Dd. ut This solution can be constructed by means of the kernel Sd.z/. Remark 1 If m D 1 the required factorization exists and can be constructed by the periodic analogue of Hilbert transform (see above). If m 2 there is no an effective algorithm for constructing the required periodic wave factorization. One can give 1. .// . / some sufﬁcient conditions, for example, supp Fd ln G Dd [ Dd . e e m Now we consider the elliptic equation (28.4) with Ad./; Bd./ 2 C.„T /.As above, one can establish the needed relationship between periodic multidimensional Riemann boundary value problem (28.7) and the corresponding integral equation in Fourier images similar to one-dimensional case [Ga81, Mu76, VaEtAl15-1] and can obtain the following result. e ./e1./ Theorem 6 If Ad Bd admit the periodic wave factorization, then the equa- m tion (28.4) has a unique solution in the space L2.hZ /. Proof Applying the discrete Fourier transform to the equation (28.4), we obtain the following integral equation with operator Sd e e e e e Ad./.SdUd/./ C Bd./.I SdUd/./ D Vd which is equivalent to certain periodic Riemann boundary value problem similar to (28.7). It was done in [Va00] for non-periodic case, and it looks the same for a periodic case. Then, according to Theorem 5, we obtain the required assertion. ut

~~Conclusion~~

The author hopes these consideration will be useful for constructing basic elements of discrete theory of elliptic pseudo-differential equations and boundary value problems on manifolds with a boundary (possibly non-smooth) taking into account latest author’s results [Va11, Va13, Va15]. 324 V.B. Vasilyev

~~References~~

[BoMa48] Bochner, S., Martin, W.T.: Several Complex Variables. Princeton University Press, Princeton, NY (1948) [Es81] Eskin, G.: Boundary Value Problems for Elliptic Pseudodifferential Equations. AMS, Providence, RI (1981) [Ga81] Gakhov, F.D.: Boundary Value Problems. Dover, Mineola, NY (1981) [MiPr86] Mikhlin, S.G., Prössdorf, S.: Singular Integral Operators. Akademie, Berlin (1986) [Mu76] Muskhelishvili, N.I.: Singular Integral Equations. North Holland, Amsterdam (1976) [Sh01] Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin/Heidelberg (2001) [Ta81] Taylor, M.E.: Pseudodifferential Operators. Princeton University Press, Princeton, NJ (1981) [Tr80] Treves, F.: Introduction to Pseudodifferential Operators and Fourier Integral Opera- tors. Springer, New York, NY (1980) [VaEtAl13] Vasilyev, A.V., Vasilyev, V.B.: Discrete singular operators and equations in a half- space. Azerb. J. Math. 3, 84–93 (2013) [VaEtAl15-1] Vasil’ev, A.V., Vasil’ev, V.B.: Periodic Riemann problem and discrete convolution equations. Differ. Equ. 51, 652–660 (2015) [VaEtAl15-2] Vasilyev, A.V., Vasilyev, V.B.: Discrete singular integrals in a half-space. In: Mityushev, V., Ruzhansky, M. (eds.) Current Trends in Analysis and Its Applications. Research Perspectives, pp. 663–670. Birkhäuser, Basel (2015) [VaEtAl15-3] Vasil’ev, A.V., Vasil’ev, V.B.: On the solvability of certain discrete equations and related estimates of discrete operators. Dokl. Math. 92, 585–589 (2015) [Va00] Vasil’ev, V.B.: Wave Factorization of Elliptic Symbols: Theory and Applications. Introduction to the Theory of Elliptic Boundary Value Problems in Non-Smooth Domains. Kluwer Academic Publishers, Dordrecht/Boston/London (2000) [Va11] Vasilyev, V.B.: Asymptotical analysis of singularities for pseudo differential equations in canonical non-smooth domains. In: Constanda, C., Harris, P. (eds.) Integral Methods in Science and Engineering, pp. 379–390. Birkhäuser, Boston, MA (2011) [Va13] Vasilyev, V.B.: Pseudo differential equations on manifolds with non-smooth boundaries. In: Pinelas, S., Došlá, Z., Došlý, O., Kloeden, P.E. (eds.) Differential and Difference Equations and Applications. Springer Proceedings in Mathematics & Statistics, vol. 47, pp. 625–637. Birkhäuser, Basel (2013) [Va15] Vasilyev, V.B.: New constructions in the theory of elliptic boundary value problems. In: Constanda, C., Kirsch, A. (eds.) Integral Methods in Science and Engineering, pp. 629–641. Birkhäuser, New York, NY (2015) [Vl07] Vladimirov, V.S.: Methods of the Theory of Functions of Many Complex Variables. Dover, Mineola, NY (2007) Chapter 29 Modes Coupling Seismic Waves and Vibrating Buildings: Existence

~~D. Volkov and S. Zheltukhin~~

~~29.1 Introduction~~

The traditional approach to evaluating seismic risk in urban areas is to consider seismic waves under ground as the only cause for motion above ground. In earlier studies, seismic wave propagation was evaluated in an initial step and in a second step impacts on man-made structures were inferred. However, observational evidence has since then suggested that when an earthquake strikes a large city, seismic activity may in turn be altered by the response of the buildings. This phenomenon is referred to as the “city-effect” and has been studied by many authors, see [KaEtAl92, ErEtAl96]. More recently, [GhetAl09], Ghergu and Ionescu have derived a model for the city effect based on the equations of solid mechanics and appropriate coupling of the different elements involved in the physical setup of the problem. They then proposed a clever way to compute a numerical solution to their system of equations. In this present paper our goal is to prove that the equations modeling the city effect introduced in [VoEtAl15 ] are solvable. There is ample numerical evidence that these coupling frequencies should exist, see [VoEtAl15 ] at least in the range of physical parameters under consideration in these numerical simulations. As far as we know, there was no mathematical proof, however, that these coupling frequencies must exist before our study [VoEtAl15-2] was published. We give here an outline of this proof. In this ground/buildings setup, following [GhetAl09] and [VoEtAl15 ], the ground is modeled to be the elastic half-space x2 >0in three dimensional space, where .x1; x2; x3/ is the space variable. We only considered the anti-plane shearing case: all displacements occur in the x3 direction and are independent of x3: see Figure 29.1.

~~D. Volkov () • S. Zheltukhin Worcester Polytechnic Institute, Worcester, MA, USA e-mail: [email protected]; [email protected]~~

~~© Springer International Publishing AG 2017 325 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5_29 326 D. Volkov and S. Zheltukhin~~

~~m1j~~

~~m0j~~

~~uj X3~~

~~X1 X2~~

~~Fig. 29.1 Anti-plane shearing model: all displacements occur in the x3 direction and are independent of x3. The buildings are modeled as springs with a mass at the top and at the bottom~~

Since in the rest of this paper we won’t be using the third direction, we set 2C x D .x1; x2/. We denote by the half plane x2 >0;x3 D 0. We refer to [GhetAl09] and [VoEtAl15 ] for a careful derivation of how given the mass density, the shear rigidity of the building and the ground, the height and the width of the building, the mass at the top, and the mass at the foundation of the building. After non-dimensionalization, we arrive at the following system of equations, assuming that the building has rescaled width 1 and is standing on the x1 axis, so that its Œ 1 ; 1 0 foundation may be assumed to be the line segment D 2 2 f g,

~~˚ C k2˚ D 0 in 2C; (29.1) ˚ D 1 on ; (29.2) @˚ D 0 on fx2 D 0gn; (29.3) @x2 @˚ ik˚ D o.r1/; uniformly as r !1 (29.4) @r Z @˚ q.k2/ D p.k2/Re .s;0/ds (29.5) @x2 where~~

p.t/ D C1t C2; q.t/ D t.C3t C C4/ (29.6) 29 Modes Coupling Seismic Waves and Vibrating Buildings: Existence 327 q 2 2 Here k >0is the wavenumber, r D x1 C x2, the rescaled physical displacement ikt is Re ˚e , and the constants C1; C2; C3; C4, are determined by the physical properties of the underground and the building as speciﬁed in [GhetAl09, VoEtAl15 ]. Note that system (29.1–29.6)isnonlinear in the unknown wavenumber k. The goal of this paper is to show the following theorem:

Theorem 1 For any positive value of the constants C1; C2; C3; C4, the system of equations (29.1–29.6) has at least one solution in k, that is, there exist a positive k and a function ˚ which has locally H1 regularity in 2C such that equations (29.1– 29.6) are satisfied. Moreover, this system of equations has at most a finite number of solutions in k. Standard arguments can show that if we fix a positive k the system of equations (29.1) through (29.4) is uniquely solvable. Theorem 1 asserts that for some of those k’s the additional relation (29.5) will hold. Here is a sketch of the proof of theorem 1.Fork in .0; 1/ we define Z @˚ F.k/ D q.k2/ p.k2/Re k .s;0/ds @x2 where ˚k solves (29.1) through (29.4). We will first show that F is real analytic in k. Then we will perform a low frequency and a high frequency analysis of ˚k.The low frequency analysis will show that F must be negative in .0; ˛/ for some positive number ˛. The high frequency analysis will prove that lim F.k/ D1, concluding k!1 the proof of theorem 1.

~~29.2 Low Frequency Analysis~~

~~1 1 We deﬁne the linear operator Tk which maps H 2 .@D/ into H 2 .@D/ by the formula: X1 in for f D ane , nD1~~

X1 H0 .k/ T .f / D a ein k n ; k n H .k/ nD1 n where Hn is the Hankel function of the ﬁrst kind of order n. Thanks to the properties of Hankel functions, we can prove 1 Lemma 1 Let f be in H 2 .@D/. Then, denoting by <; > the duality bracket between 1 1 H 2 .@D/ and H 2 .@D/ we have the following inequality:

~~Re < Tk.f /; f > Ä 0: 328 D. Volkov and S. Zheltukhin~~

Tk is the Dirichlet to Neumann operator for Helmholtz problems outside the unit disk. In fact, the following result pertaining to exterior Dirichlet problems is well known. Lemma 2 Let k >0be the wave number, D the unit disk centered at .0; 0/, and f 1 be a function in the Sobolev space H 2 .@D/. The problem

~~. C k2/u D 0 in 2 n D u D fon@D @u iku D o.r1/; uniformly as r !1 @r~~

~~X1 in has a unique solution. Writing f D ane , we have nD1~~

~~X1 H .kr/ u D a ein n ; n H .k/ nD1 n~~

~~This series and all its derivatives are uniformly convergent on any subset of 2 in the form r A where A >1. 1 Deﬁning for f in the Sobolev space H 2 .@D/~~

X1 in T0.f / D jnjane ; nD1 we can prove that Tk ! T0 in operator norm, and that T0 is the Dirichlet to Neumann operator for the Laplace operator outside the unit disk. . 1 ; 1 / 0 ˝ 2 < Recall that is the line segment 2 2 f g. Set Dfx 2 Wjxj 1 1 1gn : see Figure 29.2.LetH0; .˝/ be the closed subspace of H .˝/ consisting of functions whose upper and lower trace on are zero. Deﬁne Z Z 2 ak.u;v/D ru rv k uv .Tku/v ˝ @D

1 1 for .u;v/ in H0; .˝/ H0; .˝/.LetL be a continuous linear functional on 1 H0; .˝/. Thanks to lemma 1 and standard functional analysis arguments, the following variational problem has a unique solution for any k >0: 1 ﬁnd u in H0; .˝/ such that Z Z 2 ru rv k uv .Tku/v D L.v/; ˝ @D 29 Modes Coupling Seismic Waves and Vibrating Buildings: Existence 329

~~Fig. 29.2 , the line . 1 ; 1 / 0 segment 2 2 f g and ˝ Dfx 2 2 Wjxj <1gn~~

1 for all v in H0; .˝/.Nowlet be a smooth compactly supported function in D which is equal to 1 on and such that .x1; x2/ D .x1; x2/. For all k 0 we set 1 uQk in H0; .˝/ to be the solution to Z 2 1 ak.uQk;v/D . C k /v; 8v 2 H0; .˝/ (29.7) ˝ and we set uk DQuk C . The following directly results from the deﬁnition of uk.

Lemma 3 For k >0,uk satisﬁes the following properties: 1 (i) uk is in H .˝/. (ii) The upper and lower trace on of uk are both equal to the constant 1. 2 (iii) uk can be extended to a function in n such that, if we still denote by uk that extension, 2 2 . C k /uk D 0 in n , @ uk . 1/; @ ikuk D o r uniformly as r !1 r 2 uk.x1; x2/ D uk.x1; x2/, for all .x1; x2/ in n @u k . 1;0/ 0 .0; 1/ @x2 x D ,if x … . We are now ready to state our ﬁrst convergence result. R 2 1 Lemma 4 Let uQk be the solution to ak.uQk;v/D ˝ . C k /v; 8v 2 H0; .˝/ for all k 0, and set uk DQuk C (as previously).

~~(i) uk is analytic in k for k >0. 1 (ii) uk converges strongly to u0 in H0; .˝/. More precisely there is a constant C such that~~

~~. 2 / kuk u0kH1.˝/ Ä C k CkTk T0k~~

~~The proof of lemma 4 is to be found in [VoEtAl15-2]. Thanks to lemma 4,the following theorem was derived in [VoEtAl15-2]. 330 D. Volkov and S. Zheltukhin~~

~~R @uk H1.k/ Fig. 29.3 The decimal logs of and k against the decimal log of k. u was computed @x2 H0.k/ k by the integral equation method outlined in [GhetAl09, VoEtAl15]~~

Theorem 2 The following estimate as k approaches 0C holds Z @u H1.k/ k . /1 k ln k (29.8) @x2 H0.k/ R @uk Consequently, Re must be strictly positive for small values of k >0. @x2 Formula (29.8) can be veriﬁed numerically: uk can be computed numerically by solving an adequateR integral equation, see [VoEtAl15]. We plotted in Figure 29.3 @uk H1.k/ the decimal logs of and k . @x2 H0.k/

29.3 High Frequency Asymptotic Analysis R @uk We discuss in this section a derivation of an equivalent for as k !1, where @x2 uk DQuk C , and uQk solves variational problem (29.7). We will prove the following assertion. 29 Modes Coupling Seismic Waves and Vibrating Buildings: Existence 331

~~Theorem 3 Let uQk be the solution to problem (29.7), and set uk DQuk C . Then, as k !1, Z @ uk . 3=4/: Dik C O k (29.9) @x2~~

High frequency approximation for the wave equation is a vast subject which has been extensively studied over time. Historically, investigators have tried to explain how the laws of geometric optics relate to the wave equation at high frequency in an attempt to provide a sound foundation for Fresnel’s laws. Kirchhoff may have been the first one to write specific equations and asymptotic formulas for high frequency wave phenomena, however, his derivation was informal. A more mathematically rigorous study of the behavior of solutions to the wave equation at high frequency requires the use of Fourier integral operators and micro local analysis. As far as we know this kind of work was pioneered by Majda, Melrose, and Taylor, see [Ma76, MaEtAl77, MeEtAl85]. These authors were actually interested in the case of the exterior of a bounded convex domain, so their results cannot be applied to our case since has empty interior in 2. We have instead to rely on recent groundbreaking work by Hewett, Langdon, and Chandler-Wilde, see [ChEtAl15, ChEtAl15] which pertains to either scattering in dimension 2 by soft or hard line segments (our case), or scattering in dimension 3 by soft or hard open planar surfaces. The great achievement of their work is that they were able to derive continuity and coercivity bounds that depend explicitly on the wavenumber. Following the work by Hewett and Chandler-Wilde we introduce relevant functional spaces and frequency depending norms. Let v be a tempered distributionR on s and vO its Fourier transform. Let s be in . We say that v is in H ./ if .1 C 2/sjOv./j2d<1. We then define in Hs./ the k dependent norm Z 2 2 s 2 1=2 kvkHs./ D . .k C / jOv./j d/ : k

~~Note that H1./ is included in H1=2./ and more precisely,~~

~~v 1=2 v ; k k 1=2./ Ä k k kH1./ Hk k~~

v 1./ . 1 ; 1 / s. / for all in H . Denote by I the interval 2 2 . HQ I is defined to be the closure 1. / of Cc I (which is the space of smooth functions, compactly supported in I)forthe s s./ H .I/ I norm kkHk . is defined to be the space of restrictions to of elements in Hs./. We define on Hs.I/ the norm

~~s v s. / V s./ V H ./ V v k kHk I D inffk kHk W 2 k and jI D g (29.10)~~

Theorem 4 (Hewett and Chandler-Wilde) For any s in , the operator Sk deﬁned by the following formula for smooth functions v on I 332 D. Volkov and S. Zheltukhin Z i .Skv/.x1/ D H0.kjx1 y1j/v.y1/dy1 I 4 can be extended to a continuous linear operator from HQ s.I/ to HsC1.I/. Further- 1 more, Sk is injective, Sk is continuous, and satisﬁes the estimate p 1v 2 2 v ; kSk k Q 1=2. / Ä k k 1=2. / (29.11) Hk I Hk I

v 1=2. / for all in Hk I . Let us emphasize one more time that although the continuity and coercivity properties of Sk have been known for some time, Hewett and Chandler-Wilde’s great achievement was to derive the dependency of the coercivity and the continuity bounds on the wavenumber k as in estimate (29.11): the dependency appears in the : s. / use of the special norms k kHk I . We now sketch the derivation of asymptotic formula (29.9). Lemma 5 The following estimate holds as k approaches inﬁnity 1 . / . 1=4/ kSk ik k 1=2. / D O k (29.12) 2 Hk I

OutlineR of proof: 1 . / 1 . /. / 1 . / As 0 H0 z dz D , Sk ik x1 2 Dgk x1 , where Z Z 1 1 gk.x1/ D H0.v/dv C H0.v/dv: 1 1 k. 2 Cx1/ k. 2 x1/ R R 2 . 1=2/ 0 2 . / In [VoEtAl15-2], we showed that jQgkj D O k and jQgkj D O k ln k , . 3=4/ . 1=4/ from which it is clear that kQgkkH1./ D O k and kQgkk 1=2./ D O k . k Hk Further estimates show that Lemma 6 The following estimate holds as k approaches inﬁnity

~~1 . 1=2/ k k 1=2. / D O k (29.13) Hk I~~

~~From lemmas 5 and 6 we can now complete the proof of Theorem 3. Since . / 1 Sk fk D 2 , combining estimate (29.12) with theorem 4 it follows that~~

~~. 1=4/: kik fkk Q 1=2. / D O k Hk I~~

~~Using (29.13) we conclude that~~

~~3=4 < ik fk;1>D O.k / 29 Modes Coupling Seismic Waves and Vibrating Buildings: Existence 333~~

R R 1 @uk 1 @uk Fig. 29.4 Left: computed values of Im against k. Right: computed values of Re k @x2 k @x2 against k in other words Z @ uk . 3=4/; Dik C O k @x2 so Theorem 3 is proved. Asymptotic formula (29.9) is illustrated in Figure 29.4. We are now ready to prove Theorem 1. Recall that Z @˚ F.k/ D q.k2/ p.k2/Re k .s;0/ds @x2 where p.t/ D C1t C2, q.t/ D t.C3t C C4/, and C1; C2; C3; C4 are positive. Using 1 estimates (29.8) and (29.9) it follows that F.k/ C2 .ln k/ , k ! 0 and F.k/ 4 C3k , k !1. Since we showed that F is analytic in k, Theorem 1 is shown.

~~References~~

[ChEtAl15] Chandler-Wilde, S.N., Hewett, D.P.: Wavenumber-explicit continuity and coercivity estimates in acoustic scattering by planar screens. Integr. Equ. Oper. Theory 82, 423–449 (2015) [ChEtAl15] Chandler-Wilde, S.N., Hewett, D.P., Langdon, S., Twigger A.: A high frequency boundary element method for scattering by a class of nonconvex obstacles. Numer. Math. 129, 647–689 (2015) [ErEtAl96] Erlingsson, S., Bodare, A.: Live load induced vibrations in Ullevi Stadium - dynamic underground analysis. Undergr. Dyn. Earthq. Eng. 15, 171–188 (1996) [GhetAl09] Ghergu, M., Ionescu, I.R.: Structure-underground-structure coupling in seismic excitation and “city-effect”. Int. J. Eng. Sci. 47, 342–354 (2009) [KaEtAl92] Kanamori, H., Mori, J., Sturtevant, B., Anderson, D.L., Heaton, T.: Seismic excitation by space shuttles. Shock Waves 2, 89–96 (1992) 334 D. Volkov and S. Zheltukhin

[Ma76] Majda, A.: High frequency asymptotics for the scattering matrix and the inverse problem of acoustical scattering. Commun. Pure Appl. Math. 29, 261–291 (1976) [MaEtAl77] Majda, A., Taylor, M.E.: The asymptotic behavior of the diffraction peak in classical scattering. Commun. Pure Appl. Math. 30, 639–669 (1977) [MeEtAl85] Melrose, R.B., Taylor, M.E.: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv. Math. 55-3, 242–315 (1985) [VoEtAl15] Volkov, D., Zheltukhin, S.: Preferred frequencies for coupling of seismic waves and vibrating tall buildings. Dyn. Earthq. Eng. 74, 25–39 (2015) [VoEtAl15-2] Volkov, D., Zheltukhin, S.: Existence of frequency modes coupling seismic waves and vibrating tall buildings. J. Math. Anal. Appl. 421, 276–297 (2015) Index

A boundary-domain integral equation system, abstract Cauchy problem associated with the boundary-domain integral equations, generator of a Markov semigroup, 215 12 Boundary-Domain Integral Equations, 223 additive perturbation, 16, 17 anisotropic media, 139 approximation formula for .T.t//t0, 12 C approximation formula for solutions to canonical elliptic second-order differential evolution problems, 12 operator associated with a Markov approximation process, 11 operator, 10, 11 approximation process generated by a Markov canonical projection on the simplex, 14 operator, 9 Caputo fractional derivatives, 107 asymptotic coefﬁcients of WT , 11 solutions, 259 conjugate differential forms, 61 asymptotic behavior of a Markov semigroup, core, 12 18 asymptotic formula concerned with a second-order differential D operator, for Bernstein-Schnabl diffeomorph conformal methodology, 205 operators, 11 diffeomorph conformal transformation, 205 Asymptotic solutions, 259 differential operator generated by a Markov operator, 9 Dirichlet boundary value problem, 293 B Dirichlet problem, 184 barycenter of a probability Borel measure, 17 Dirichlet to Neumann operator, 327 BDIES, 218 discreteness, periodicity, 315 Bernstein-Schnabl operator associated with a discreteness, periodicity, holomorphy, and Markov operator, 10 factorization, 315 biharmonic equation, 67 domain bilateral reduction, 67 multiply connected, 185 boundary integral equations, 139 domains boundary integral equations method two-component, 83 indirect, 185 double boundary-domain integral equation system, integral 218 transforms, 277

© Springer International Publishing AG 2017 335 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Volume 1, DOI 10.1007/978-3-319-59384-5 336 Index double Laplace transform and fractional integral laplacian, 277 equations, 1, 71, 183 double-layer potential, 59 equations, nonuniqueness, 35 equations, solvability, 47 integral equation of the first kind, 60 E interface jump condition, 83 elastic plates, 71 interior transmission eigenvalues, 139, 149 elasticity, 183 inverse scattering, 149 elastostatics, 183 iterate of an operator, 12 ellipsoid, 13 equations quasilinear elliptic, 83 L singular, 83 Laplace equation, 293 Lebedev quadrature, 149 locally convex Hausdorff space, 17 F Lototsky-Schnabl operator, 16 far-field operator, 149 Favard class, 18 flow problems, complex boundary geometries, M 205 Markov fractional operator, 9, 10 laplacian, 277 projection, 13 Fractional circle Zernike polynomials, 265 semigroup, 12 Fredholmness of nonlocal singular integral Maxwell operators, 95 equations, 259 function Maxwell equations, 259 ith coordinate, 11 modified Bernstein-Schnabl operator, 17 constant, 10 multiple-layer potentials, 65 continuous, 10 multiplicative perturbation, 16 convex, 13 Muskhelishvili’s Method, 61 Hölder continuous, 13 fundamental solution, 107 N G n-Laplacian, 119 Galerkin–Petrov boundary element method, Neutron space kinetics, 249 293 neutron space kinetics, cylinder symmetry, 249 generalized Fourier series method, 71 New, 215 generalized Kantorovich operator, 16 nonlinear eigenvalue problem, 139

H O Hankel functions, 327 On a class of integral equations involving heuristic convergence, 173 kernels of cosine and sine type, 47 Hill equation, 193 On the radiative conductive transfer equation, hinged plates, 35 173 holomorphy, factorization, 315 operator homogenization Bernstein, 11 Robin boundary conditions, 119 Bernstein-Schnabl, 10 hypercube, 16 Fleming-Viot , 14 generalized Kantorovich, 16 Laplace, 14 I linear positive, 10 initial-boundary value evolution problem, 12 Lototsky-Schnabl, 16 inside-outside duality method, 149 Markov, 10 Index 337 operator (cont.) set of interpolation points, 11 modiﬁed Bernstein-Schnabl, 17 shift, Wiener Poisson, 13 type optimal design, 161 algebra, invertibility, 95 Orlicz–Sobolev spaces, 161 simple-layer potential, 59 simplex, 14 single layer potential, 293 P spatial regularity properties for evolution parametrix, 215, 219 equations, 12 Parseval spherical t-Design, 149 identity, convolution, 47 stability analysis, 293 perforated domains Steady-state crack propagation, 237 homogenization, 119 strongly admissible sequence for .T.t//t0, 12 perimeter, 161 suspension bridges, 193 plane elasticity, 35, 183 Poincaré map, 193 T potential tau method for nonlinear integro-differential elastic double-layer, 184 equations, 305 elastic single-layer, 184 tensor product of Markov operators, 15 power increase, 249 Theorem preservation properties of Bernstein-Schnabl Lyapunov–Tauber, 188 operators, 12 thermoelasticity, 107 probability Borel measure, 10 thin structure, 161 product time fractional diffusion operator, 107 integration, 1 torsional instability, 193 product Gaussian quadrature, 149 torsional performances, 35 product-integration method in Astrophysics, 1 Traction problem, 184 Trotter-Schnabl-type theorem, 12 R radiative conductive transfer equation, 173 Robin boundary conditions, 119 V vibrating buildings, 325 S saturation class, 18 semigroup W associated with Bernstein-Schnabl waves operators, Markov, positive, 12 seismic, 325