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Brezzi F., at Al. (Eds.) Analysis and Numerics of Partial Differential Springer INdAM Series Volum e 4 Editor-in-Chief V. Ancona Series Editors P. Cannarsa C. Canuto G. Coletti P. Marcellini G. Patrizio T. Ruggeri E. Strickland A. Verra For further volumes: www.springer.com/series/10283 Franco Brezzi r Piero Colli Franzone r Ugo Gianazza r Gianni Gilardi Editors Analysis and Numerics of Partial Differential Equations Editors Franco Brezzi Ugo Gianazza Istituto di Matematica Applicata e Dipartimento di Matematica “F. Casorati” Tecnologie Informatiche (IMATI) Università degli Studi di Pavia CNR Pavia Pavia, Italy Pavia, Italy Piero Colli Franzone Gianni Gilardi Dipartimento di Matematica “F. Casorati” Dipartimento di Matematica “F. Casorati” Università degli Studi di Pavia Università degli Studi di Pavia Pavia, Italy Pavia, Italy ISSN 2281-518X ISSN 2281-5198 (electronic) Springer INdAM Series ISBN 978-88-470-2591-2 ISBN 978-88-470-2592-9 (eBook) DOI 10.1007/978-88-470-2592-9 Springer Milan Heidelberg New York Dordrecht London Library of Congress Control Number: 2012951305 © Springer-Verlag Italia 2013 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of pub- lication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) In memory of Enrico Magenes Enrico Magenes (courtesy of the family) Preface On November 2nd 2010 Enrico Magenes passed away. One year later, some of his friends, collaborators and former students organized a three-day conference, in order to celebrate his memory and give a first assessment of his deep influence on contemporary Mathematics. All the speakers were experts in the analysis and numerics of partial differential equations, who had directly inter- acted with Magenes, during his long career. The present volume is a direct offshoot of that meeting, and it collects the main contributions offered in that occasion, properly revised and expanded. It consists of two parts: the first one gives a wide historical perspective of Magenes’ work; the second one contains original research or survey papers, and shows how ideas, methods, and techniques introduced by Magenes and his collaborators still have an impact on the current research in Mathematics. As agreed between Springer and UMI (Unione Matematica Italiana), some of the papers appearing in the second part will be fully published on Bollettino UMI as well. Although it is still too early to fully appreciate Magenes’ legacy, nonetheless the volume is a first attempt to present a comprehensive survey of his activity in Mathematics. At the same time, from Magenes’ peculiar point of view, it is a broad perspective of the research in partial differential equations and their applications developed in Italy in the period 1950–2000. The editors are grateful to Francesca Bonadei of Springer Italy for the unique opportunity offered with the publication of this volume, and for her constant support during its preparation. Pavia, Italy Franco Brezzi Piero Colli Franzone Ugo Gianazza Gianni Gilardi vii Contents Part I A Historical Perspective Personal Memories ............................... 3 Franco Brezzi Some Aspects of the Research of Enrico Magenes in Partial Differential Equations ................................. 5 Giuseppe Geymonat Enrico Magenes and the Dam Problem ................... 13 Claudio Baiocchi Inverse Problems in Electrocardiology .................... 19 Piero Colli Franzone Stefan Problems and Numerical Analysis .................. 37 Claudio Verdi Enrico Magenes and the Teaching of Mathematics ............. 47 Mario Ferrari List of Mathematical Works Authored or Edited by Enrico Magenes ... 53 Ugo Gianazza Part II Recent Developments Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case .................. 63 Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré Spaces of Finite Element Differential Forms .................117 Douglas N. Arnold A Priori Bounds for Solutions of a Nonlocal Evolution PDE ........141 Luis Caffarelli and Enrico Valdinoci ix x Contents On the Numerical Analysis of Adaptive Spectral/hp Methods for Elliptic Problems .................................165 Claudio Canuto and Marco Verani A Theory and Challenges for Coarsening in Microstructure ........193 Katayun Barmak, Eva Eggeling, Maria Emelianenko, Yekaterina Epshteyn, David Kinderlehrer, Richard Sharp, and Shlomo Ta’asan A Generalized Empirical Interpolation Method: Application of Reduced Basis Techniques to Data Assimilation .................221 Yvon Maday and Olga Mula Analysis and Numerics of Some Fractal Boundary Value Problems ....237 Umberto Mosco AFEM for Geometric PDE: The Laplace-Beltrami Operator .......257 Andrea Bonito, J. Manuel Cascón, Pedro Morin, and Ricardo H. Nochetto Generalized Reduced Basis Methods and n-Width Estimates for the Approximation of the Solution Manifold of Parametric PDEs ....307 Toni Lassila, Andrea Manzoni, Alfio Quarteroni, and Gianluigi Rozza Variational Formulation of Phase Transitions with Glass Formation ...331 Augusto Visintin List of Contributors Luigi Ambrosio Scuola Normale Superiore, Pisa, Italy Douglas N. Arnold School of Mathematics, University of Minnesota, Minneapolis, MN, USA Claudio Baiocchi Dipartimento di Matematica “F. Casorati”, Università di Pavia, Pavia, Italy; Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Rome, Italy Katayun Barmak Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, USA Andrea Bonito Department of Mathematics, Texas A&M University, College Sta- tion, TX, USA Franco Brezzi Istituto di Matematica Applicata e Tecnologie Informatiche “E. Ma- genes”, Consiglio Nazionale delle Ricerche, Pavia, Italy Luis Caffarelli Department of Mathematics, University of Texas at Austin, Austin, TX, USA Claudio Canuto Dipartimento di Scienze Matematiche, Politecnico di Torino, Turin, Italy J. Manuel Cascón Departamento de Economía e Historia Económica, Universidad de Salamanca, Salamanca, Spain Piero Colli Franzone Dipartimento di Matematica “F. Casorati”, Università di Pavia, Pavia, Italy Eva Eggeling Visual Computing, Fraunhofer Austria Research GmbH, Graz, Aus- tria Maria Emelianenko Department of Mathematical Sciences, George Mason Uni- versity, Fairfax, VA, USA xi xii List of Contributors Yekaterina Epshteyn Department of Mathematics, The University of Utah, Salt Lake City, UT, USA Mario Ferrari Dipartimento di Matematica “F. Casorati”, Università di Pavia, Pavia, Italy Giuseppe Geymonat LMS, Laboratoire de Mécanique des Solides UMR 7649, École Polytechnique, Palaiseau Cedex, France Ugo Gianazza Dipartimento di Matematica “F. Casorati”, Università di Pavia, Pavia, Italy Nicola Gigli Université de Nice, Mathématiques, Nice, France David Kinderlehrer Department of Mathematical Sciences, Carnegie Mellon Uni- versity, Pittsburgh, PA, USA Toni Lassila MATHICSE-CMCS, Modelling and Scientific Computing, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland Yvon Maday Laboratoire Jacques-Louis Lions, UMR 7598, UPMC Univ Paris 06, Paris, France; Division of Applied Mathematics, Institut Universitaire de France and Brown University, Providence, RI, USA Andrea Manzoni SISSA Mathlab, SISSA—International School for Advanced Studies, Trieste, Italy Pedro Morin Instituto de Matemática Aplicada del Litoral (IMAL), Güemes 3450 and Departamento de Matemática, Facultad de Ingeniería Química, Universidad Na- cional del Litoral, Santa Fe, Argentina Umberto Mosco Department of Mathematical Sciences, Worcester Polytechnic In- stitute, Worcester, MA, USA Olga Mula Laboratoire Jacques-Louis Lions, UMR 7598, UPMC Univ Paris 06, Paris, France; CEA Saclay—DEN/DANS/DM2S/SERMA/LLPR, Gif-Sur-Yvette Cedex, France Ricardo H. Nochetto Department of Mathematics and Institute for Physical Sci- ence and Technology, University of Maryland, College Park, MD, USA Alfio Quarteroni MATHICSE-CMCS, Modelling and Scientific Computing, École
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