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Analysis, Partial Differential Equations and Applications: The Operator Theory: Advances and Applications Vol. 193 Editor: I. Gohberg Editorial Office: School of Mathematical Sciences V. Olshevski (Storrs, CT, USA) Tel Aviv University M. Putinar (Santa Barbara, CA, USA) Ramat Aviv A.C.M. Ran (Amsterdam, The Netherlands) Israel L. Rodman (Williamsburg, VA, USA) J. Rovnyak (Charlottesville, VA, USA) B.-W. Schulze (Potsdam, Germany) F. Speck (Lisboa, Portugal) Editorial Board: I.M. Spitkovsky (Williamsburg, VA, USA) D. Alpay (Beer Sheva, Israel) S. Treil (Providence, RI, USA) J. Arazy (Haifa, Israel) C. Tretter (Bern, Switzerland) A. Atzmon (Tel Aviv, Israel) H. Upmeier (Marburg, Germany) J.A. Ball (Blacksburg, VA, USA) N. Vasilevski (Mexico, D.F., Mexico) H. Bart (Rotterdam, The Netherlands) S. Verduyn Lunel (Leiden, The Netherlands) A. Ben-Artzi (Tel Aviv, Israel) D. Voiculescu (Berkeley, CA, USA) H. Bercovici (Bloomington, IN, USA) D. Xia (Nashville, TN, USA) A. Böttcher (Chemnitz, Germany) D. Yafaev (Rennes, France) K. Clancey (Athens, GA, USA) R. Curto (Iowa, IA, USA) K. R. Davidson (Waterloo, ON, Canada) Honorary and Advisory Editorial Board: M. Demuth (Clausthal-Zellerfeld, Germany) L.A. Coburn (Buffalo, NY, USA) A. Dijksma (Groningen, The Netherlands) H. Dym (Rehovot, Israel) R. G. Douglas (College Station, TX, USA) C. Foias (College Station, TX, USA) R. Duduchava (Tbilisi, Georgia) J.W. Helton (San Diego, CA, USA) A. Ferreira dos Santos (Lisboa, Portugal) T. Kailath (Stanford, CA, USA) A.E. Frazho (West Lafayette, IN, USA) M.A. Kaashoek (Amsterdam, The Netherlands) P.A. Fuhrmann (Beer Sheva, Israel) P. Lancaster (Calgary, AB, Canada) B. Gramsch (Mainz, Germany) H. Langer (Vienna, Austria) H.G. Kaper (Argonne, IL, USA) P.D. Lax (New York, NY, USA) S.T. Kuroda (Tokyo, Japan) D. Sarason (Berkeley, CA, USA) L.E. Lerer (Haifa, Israel) B. Silbermann (Chemnitz, Germany) B. Mityagin (Columbus, OH, USA) H. Widom (Santa Cruz, CA, USA) Analysis, Partial Differential Equations and Applications The Vladimir Maz’ya Anniversary Volume Alberto Cialdea Flavia Lanzara Paolo Emilio Ricci Editors Birkhäuser Basel · Boston · Berlin Authors: Alberto Cialdea Flavia Lanzara Dipartimento di Matematica e Dipartimento di Matematico “Guido Castelnuovo” Informatica Sapienza Università di Roma Università della Basilicata Piazzale Aldo Moro 2 Viale dell’Ateneo Lucano 10 00185 Rome, Italy 85100 Potenza, Italy e-mail: [email protected] e-mail: [email protected] Paolo Emilio Ricci Dipartimento di Matematico “Guido Castelnuovo” Sapienza Università di Roma Piazzale Aldo Moro 2 00185 Rome, Italy e-mail: [email protected] 2000 Mathematical Subject Classification: 31-06, 35-06, 46-06 Library of Congress Control Number: 2009931267 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de ISBN 978-3-7643-9897-2 Birkhäuser Verlag AG, Basel - Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-7643-9897-2 e-ISBN 978-3-7643-9898-9 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents A. Cialdea, F. Lanzara and P.E. Ricci On the Occasion of the 70th Birthday of Vladimir Maz’ya . ix Yu. Burago and N.N. Kosovsky Boundary Trace for BV Functions in Regions with Irregular Boundary . 1 J. Burnett, O. Chervova and D. Vassiliev DiracEquationasa SpecialCaseofCosseratElasticity .............. 15 I. Capuzzo Dolcetta H¨older and Lipschitz Estimates for Viscosity Solutions of Some Degenerate Elliptic PDE’s . 31 A. Cialdea Criteria for the Lp-dissipativity of Partial Differential Operators . 41 A. Cianchi SharpEstimatesforNonlinearPotentialsandApplications ........... 57 M. Frazier and I. Verbitsky Solvability Conditions for a Discrete Model of Schr¨odinger’sEquation ............................................ 65 Yu.I. Karlovich An Algebra of Shift-invariant Singular Integral Operators with Slowly Oscillating Data and Its Application to Operators witha CarlemanShift ............................................... 81 C.O. Kiselman FrozenHistory:ReconstructingtheClimateofthePast .............. 97 G. Kresin Multidimensional Harmonic Functions Analogues of Sharp Real-partTheoremsinComplexFunctionTheory .................... 115 vi Contents F. Lanzara and G. Schmidt Cubature of Integral Operators by Approximate Quasi-interpolation .................................................. 129 S. Mayboroda and V. Maz’ya Pointwise Estimates for the Polyharmonic Green Function inGeneralDomains ................................................. 143 R. McOwen On Elliptic Operators in Nondivergence and in Double DivergenceForm .................................................... 159 D. Mitrea and M. Mitrea On the Well-posedness of the Dirichlet Problem in Certain Classes of Nontangentially Accessible Domains . 171 S. Molchanov and B. Vainberg On Negative Spectrum of Schr¨odingerTypeOperators ............... 197 G. Moscariello, A. Passarelli di Napoli and C. Sbordone ACL-homeomorphismsinthePlane .................................. 215 D. Natroshvili and Z. Tediashvili CrackProblemsforCompositeStructures ............................ 227 Y. Pinchover and K. Tintarev On Positive Solutions of p-Laplacian-typeEquations ................. 245 J. Rossmann Mixed Boundary Value Problems for Stokes and Navier-StokesSystemsinPolyhedralDomains ....................... 269 S. Samko On Some Classical Operators of Variable Order inVariableExponentSpaces ......................................... 281 M.A. Vivaldi Irregular Conductive Layers . 303 W.L. Wendland OntheDoubleLayerPotential ...................................... 319 Vladimir Maz’ya Operator Theory: Advances and Applications, Vol. 193, ix–xvii c 2009 Birkh¨auser Verlag Basel/Switzerland On the Occasion of the 70th Birthday of Vladimir Maz’ya Alberto Cialdea, Flavia Lanzara and Paolo E. Ricci This volume includes a selection of lectures given at the International Workshop “Analysis, Partial Differential Equations and Applications”, held at the Mathe- matical Department of Sapienza University (Rome, June 30th–July 3rd, 2008), on the occasion of the 70th birthday of Vladimir Maz’ya. Besides Italy, twenty seven countries were represented there: Belarus, Canada, China, Colombia, Croatia, Czech Republic, Finland, France, Georgia, Germany, Greece, Israel, Mexico, New Zealand, Poland, Portugal, Rumania, Russia, Saudi Arabia, South Korea, Spain, Sweden, Taiwan, The Netherlands, Turkey, United Kingdom, and United States of America. It is not surprising that the decision of the Italian National Institute for Advanced Mathematics “F. Severi” (INDAM) to dedicate a Workshop to Vladimir Maz’ya was crowned by such great success. The scientific and human endowments of Maz’ya are well known. He has inspired numerous researchers in Analysis and its applications, among them many in Italy. Maz’ya gladly acknowledges that this inspiration has been mu- tual. The Italian school of Analysis and PDEs has played an important role in his development, starting with his undergraduate years 1955–1960 and continuing to this day. As a third year student, through S. Mikhlin’s lectures, he became ac- quainted with Tricomi’s pioneering work on multi-dimensional singular integrals [37], [38], a topic of Maz’ya’s keen interest in the future ([14], [21] and others). A year later, Vladimir discovered the equivalence of various Sobolev type inequal- ities with isoperimetric and isocapacitary inequalities, which strongly influenced functional analysis and partial differential equations in subsequent years. In par- ticular, he found the sharp constant in the E. Gagliardo inequality between the Ln/(n−1) norm of a function and the L1 norm of its gradient [10]. Later Gagliardo’s results on boundary traces of Sobolev functions were developed by Maz’ya and his colleagues in various directions (see, for example, [24], [33], [34]). Following Mikhlin’s recommendation, Maz’ya read the Russian translation of Carlo Miranda’s “Equazioni alle Derivate Parziali di Tipo Ellittico” [36], which had appeared in 1957 in Moscow. This comprehensive survey of the Italian contribution x A. Cialdea, F. Lanzara and P.E. Ricci to the field, which at that time was undergoing a major expansion, became the first book on PDEs to be read by the young Maz’ya. Miranda’s book strongly influenced the shaping of Vladimir’s professional interests. An evidence to this is his first publication which appeared in [9] exactly 50 years ago. The year 1957 saw the appearance of the seminal article by E. De Giorgi on the H¨older regularity of solutions to elliptic second-order equations with measur- able bounded coefficients, which had a tremendous impact on the theory of PDEs, not least the work of Maz’ya. In the article [11] of 1961, he solved a problem posed by G. Stampacchia on an estimate of weak solutions to the equations just men- tioned. One of the original traits of this short paper was a characterization of the boundary in terms of an isoperimetric function introduced by the author, which
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