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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 764937-89-839-7-2 e-ISBN 764937-89-839-8-9 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 Dirac Equation as a Special Case of Cosserat Elasticity .............. 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 PartialDifferential Operators ..... 41 A. Cianchi Sharp Estimates for Nonlinear Potentials and Applications ........... 57 M. Frazier and I. Verbitsky Solvability Conditions for a Discrete Model of Schro¨dinger’s Equation ............................................ 65 Yu.I. Karlovich An Algebra of Shift-invariant Singular Integral Operators with Slowly Oscillating Data and Its Application to Operators with a Carleman Shift ............................................... 81 C.O. Kiselman Frozen History: Reconstructing the Climate of the Past .............. 97 G. Kresin Multidimensional Harmonic Functions Analogues of Sharp Real-part Theorems in Complex Function Theory .................... 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 in General Domains ................................................. 143 R. McOwen On Elliptic Operators in Nondivergence and in Double Divergence Form .................................................... 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 Schro¨dinger Type Operators ............... 197 G. Moscariello, A. Passarelli di Napoli and C. Sbordone ACL-homeomorphisms in the Plane .................................. 215 D. Natroshvili and Z. Tediashvili Crack Problems for Composite Structures ............................ 227 Y. Pinchover and K. Tintarev On Positive Solutions of p-Laplacian-type Equations ................. 245 J. Rossmann Mixed Boundary Value Problems for Stokes and Navier-Stokes Systems in Polyhedral Domains ....................... 269 S. Samko On Some Classical Operators of Variable Order in Variable Exponent Spaces ......................................... 281 M.A. Vivaldi Irregular Conductive Layers ......................................... 303 W.L. Wendland On the Double Layer Potential ...................................... 319 Vladimir Maz’ya OperatorTheory: Advances andApplications,Vol.193, ix–xvii (cid:2)c 2009Birkh¨auserVerlagBasel/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- maticalDepartmentofSapienzaUniversity(Rome,June 30th–July3rd,2008),on the occasion of the 70th birthday of Vladimir Maz’ya. BesidesItaly,twentysevencountrieswererepresentedthere: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 AdvancedMathematics“F.Severi”(INDAM)todedicateaWorkshoptoVladimir Maz’yawascrownedbysuchgreatsuccess.Thescientific andhumanendowments of Maz’ya are well known. HehasinspirednumerousresearchersinAnalysisanditsapplications,among themmanyinItaly.Maz’yagladlyacknowledgesthatthisinspirationhasbeenmu- tual. The ItalianschoolofAnalysis and PDEshas playedanimportantrole in his development, starting with his undergraduate years 1955–1960and 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) normofafunctionandtheL1normofitsgradient[10].LaterGagliardo’s resultsonboundarytracesofSobolevfunctionsweredevelopedbyMaz’yaandhis colleagues in various directions (see, for example, [24], [33], [34]). FollowingMikhlin’srecommendation,Maz’yareadtheRussiantranslationof CarloMiranda’s“EquazionialleDerivateParzialidiTipoEllittico”[36],whichhad appearedin1957inMoscow.ThiscomprehensivesurveyoftheItaliancontribution 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- ableboundedcoefficients,whichhadatremendousimpactonthetheoryofPDEs, notleasttheworkofMaz’ya.Inthearticle[11]of1961,hesolvedaproblemposed 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 characterizationof the boundary in terms of an isoperimetric function introduced by the author, which enabledhimtostudythesharpdependenceoftheregularitypropertiesofsolutions tothe Neumannproblemonthe behaviourofthe boundary.Adetailedexposition of this work, containing a wealth of new ideas, was published in [18], 1969. In[12],1963,Maz’yaobtainedhisfamousestimateofthecontinuitymodulus of a solutionto the Dirichlet problemnear a boundary point, formulatedin terms of the Wiener integral (see also [15], [16]). Later, a result of the same nature was obtained by him for nonlinear equations including the p-Laplacian [19]. It is noteworthy that the classical paper by Littman, Stampacchia and Weinberger [8] on the Wiener regularity of a boundary point was translated into Russian by Maz’ya for the Moscow collection of translations “Matematika” from a preprint, even before its publication in a journal. Of exceptional importance were Maz’ya’s counterexamples relating to the 19thand20thHilbertproblemsforhigher-orderellipticequationswhichappeared in[17],1968,independentlyofandsimultaneouslywithanalogouscounterexamples of E. De Giorgi and E. Giusti–M.Miranda. The results of L. Cesari, R. Caccioppoli and especially E. De Giorgi on gen- eralization of the notion of the surface area on nonsmooth surfaces played an important role in the pioneering research of Maz’ya and his coauthors in the the- oryofharmonicpotentialsonnonsmoothdomainsaswellasinthetheoryofspaces of functions with bounded variation [1], [13], [2], [3]. The influence of G. Cimmino’s results of 1937 [4] on the Dirichlet problem withboundarydatainL aswellasG.Fichera’sunifiedtheoryofelliptic-parabolic p equations [5] can be traced in Maz’ya’s breakthrough work on the generic degen- erating oblique derivative problem [20]. Oneofthefundamentalresultsinthe theoryofpartialdifferentialequations, theC.Miranda–Sh.Agmonmaximumprincipleforhigher-orderellipticequations, was crucially developed by Maz’ya and his collaborators in several directions: polyhedral domains [23], sharp constants [22], parabolic systems [7]. The above, by necessity a rather incomplete survey, clearly shows that the ItalianschoolstimulatedtheearlyworkofMaz’yainspiteoftheironcurtain.With time the contacts became bilateral and even personal. At the moment, Maz’ya is collaborating with a number of Italian mathematicians which can be seen, for instance, in some papers included into the present volume.

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Birkhäuser Basel, 2009. - 352 pages.This volume includes several invited lectures given at the International Workshop "Analysis, Partial Differential Equations and Applications", held at the Mathematical Department of Sapienza University of Rome, on the occasion of the 70th birthday of Vladimir G.
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