Recent Trends in Operator Theory and Partial Differential Equations: The Roland Duduchava Anniversary Volume PDF
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Operator Theory Advances and Applications 258 Vladimir Maz'ya David Natroshvili Eugene Shargorodsky Wolfgang L. Wendland Editors Recent Trends in Operator Theory and Partial Differential Equations The Roland Duduchava Anniversary Volume Operator Theory: Advances and Applications Volume 258 Founded in 1979 by Israel Gohberg Editors: Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Wien, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Honorary and Advisory Editorial Board: Vadim Adamyan (Odessa, Ukraine) Lewis A. Coburn (Buffalo, NY, USA) Wolfgang Arendt (Ulm, Germany) Ciprian Foias (College Station, TX, USA) Albrecht Böttcher (Chemnitz, Germany) J.William Helton (San Diego, CA, USA) B. Malcolm Brown (Cardiff, UK) Thomas Kailath (Stanford, CA, USA) Raul Curto (Iowa, IA, USA) Peter Lancaster (Calgary, Canada) Fritz Gesztesy (Columbia, MO, USA) Peter D. Lax (New York, NY, USA) Pavel Kurasov (Stockholm, Sweden) Donald Sarason (Berkeley, CA, USA) Vern Paulsen (Houston, TX, USA) Bernd Silbermann (Chemnitz, Germany) Mihai Putinar (Santa Barbara, CA, USA) Harold Widom (Santa Cruz, CA, USA) Ilya M. Spitkovsky (Williamsburg, VA, USA) Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Orange, CA, USA) Birgit Jacob (Wuppertal, Germany) André C.M. Ran (Amsterdam, The Netherlands) Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan) Ingo Witt (Göttingen, Germany) Moreinformationaboutthisseriesathttp://www.springer.com/series/4850 Vladimir Maz’ya • David Natroshvili Eugene Shargorodsky • Wolfgang L. Wendland Editors Recent Trends in Operator Theory and Partial Differential Equations The Roland Duduchava Anniversary Volume Editors Vladimir Maz’ya David Natroshvili Department of Mathematics Department of Mathematics Linköping University Georgian Technical University Linköping, Sweden Tbilisi, Georgia Eugene Shargorodsky Wolfgang L. Wendland Department of Mathematics Institut für Angewandte Analysis King’s College London und Numerische Simulation London, United Kingdom Fachbereich Mathematik Universität Stuttgart Stuttgart, Germany ISSN 0255-0156 ISSN22 96-4878 (electronic) Operator Theory: Advances and Applications ISBN 978-3-319-47077-1 ISBN 978-3-319-47079-5 (eBook) DOI 10.1007/978-3-319-47079-5 Library of Congress Control Number: 2016963805 Mathematics Subject Classification (2010): 47B35, 45E10, 45E05, 47A68, 47G30, 35S05, 35P05, 35J15, 35J25, 35J47, 35J57, 35Q74, 74B05, 74A45, 35J10, 45L05, 65R20 © 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 Contents V. Maz’ya, D. Natroshvili, E. Shargorodsky and W.L. Wendland Roland Duduchava .................................................. vii A. Bo¨ttcher The Duduchava–Roch Formula ...................................... 1 L.P. Castro and F.-O. Speck Convolution Type Operators with Symmetry in Bessel Potential Spaces ............................................. 21 S.N. Chandler-Wilde and R. Hagger On Symmetries of the Feinberg–Zee Random Hopping Matrix ....... 51 M. Costabel Inequalities of Babuˇska–Aziz and Friedrichs–Velte for Differential Forms ............................................... 79 M. Chaussade-Beaudouin, M. Dauge, E. Faou and Z. Yosibash High Frequency Oscillations of First Eigenmodes in Axisymmetric Shells as the Thickness Tends to Zero ............................... 89 V.D. Didenko and B. Silbermann Kernels of Wiener–Hopf plus Hankel Operators with Matching Generating Functions ..................................... 111 V.D. Didenko and A.M. Vu Spline Galerkin Methods for the Double Layer Potential Equations on Contours with Corners ................................ 129 E. Espinoza-Loyola and Yu.I. Karlovich C∗-algebras of Bergman Type Operators with Piecewise Continuous Coefficients over Bounded Polygonal Domains ........... 145 V.A. Kovtunenko and A.V. Zubkova Solvability and Lyapunov Stability of a Two-component System of Generalized Poisson–Nernst–PlanckEquations .................... 173 vi Contents M. Dalla Riva, M. Lanza de Cristoforis and P. Musolino A Local Uniqueness Result for a Quasi-linear Heat Transmission Problem in a Periodic Two-phase Dilute Composite ................. 193 V. Rabinovich The Method of Potential Operators for Anisotropic Helmholtz Operators on Domains with Smooth Unbounded Boundaries ........ 229 E. Cordero, F. Nicola and L. Rodino Gabor Analysis for Schr¨odinger Equations and Propagation of Singularities ...................................................... 257 A. Sa´nchez-Nungaray and N. Vasilevski Commutative Algebras of Toeplitz Operators on a Siegel Domain Associated with the Nilpotent Group of Its Biholomorphisms ........ 275 OperatorTheory: Advances andApplications,Vol.258,vii–xviii (cid:2)c 2017SpringerInternational Publishing Roland Duduchava Vladimir Maz’ya, David Natroshvili, Eugene Shargorodsky and Wolfgang L. Wendland Roland was born on 12 November 1945 in Tbilissi, Georgia. His mother was a Physics and Mathematics teacher. His father was an energy engineer who spent most of his career working in various high positions in the government of the Autonomous Republic of Abkhazia. Roland’s father was also a writer who pub- lishedthree books.He wasburied atthe Pantheonof writersandpublic figures in Sokhumi, Georgia. In 1948, Roland’s father graduated from the Georgian Polytechnic Institute and got a job at the construction of the Sokhumi power station, so the family moved from Tbilissi to Abkhazia. Roland studied at Sokhumi school No. 5 in 1952–1962 and then at the De- partmentofMechanicsandMathematicsofTbilissiStateUniversityin1962–1968. During hisuniversityyears,Rolandwasthe recipientofthe A. Razmadzescholar- ship for an outstanding student. After graduatingwith Distinction, he started his PhDstudyatA.RazmadzeMathematicalInstituteinTbilissi.Roland’ssupervisor Boris Khvedelidze arranged for him to spend most of his PhD programme at the Institute ofMathematicsofthe MoldavianAcademyofSciencesinChi¸sin˘au(then Kishinev) under the supervision of Israel Gohberg. The years spent in Kishinev have had a profound influence on Roland as a mathematician and as a person. It was then that he met his future wife Efrosinia (Zhenya) Khomenko whom he married in 1970 and with whom he has two children. RolandreturnedtoTbilissiin1971,receivedaPhDfromA.RazmadzeMath- ematical Institute and was appointed a Junior Research Fellow at the same in- stitute. Roland has stayed at the institute since then and has been a Principal Research Fellow and the Head of the Department of Mathematical Physics there since 1996. He has also held various positions at Tbilissi State University and the IB German-Georgian University in Tbilissi. Roland received a Higher Doctorate (Habilitation) from Moscow State University in 1983. In 1979, Roland received an Alexander v. Humboldt fellowship (at Erhard Meister’s initiative), whichprovidedvisiting professorshipsatthe TH Darmstadt, the University of Stuttgart, and the TU Chemnitz. He was an Invited Guest Pro- viii V. Maz’ya, D. Natroshvili, E. Shargorodsky and W.L. Wendland fessorat the Humboldt UniversityofBerlinin 1993–1994andat the University of Stuttgartin2001–2002.HeplayedaleadingrˆoleintheGeorgian-Germanresearch cooperationprojectoftheGermanResearchFoundationDFGin1994–1998.Heob- tainedaMercatorguestprofessorshipattheUniversityofSaarlandinSaarbru¨cken in 2002–2003and continued there as a Professor in 2004–2007. RolandhasservedasthePresidentoftheGeorgianMathematicalUniondur- ing 1997–2001 and since 2009. He is a member of the editorial boards of Integral Equations and Operator Theory, Journal of Applied Mathematics and Bioinfor- matics, Memoirs on Differential Equations and Mathematical Physics, Georgian Mathematical Journal, and Tbilissi Mathematical Journal. Roland’s first papers ([1–4, 8, 10]) were devoted to one-dimensional singular integral operators on weighted H¨older spaces. He completed the classical theory developedbyF.D.Gakhov,N.I.Muskhelishvili,andtheircollaborators,andrecast itintermsofoperatorsonBanachspaces.ThosepaperswerefollowedbyRoland’s well-known works on discrete Wiener–Hopf ([5–7, 14, 20, 27]), bisingular ([17, 21, 23, 31, 32]), and multidimensional convolution ([13, 17, 22, 34]) operators. Roland’sgroundbreakingworkonWiener–Hopfoperatorswithdiscontinuous symbols and integral equations with fixed singularities ([M1, M2], [9, 16, 18, 19, 26, 28–30])deserves a special mention. This tour de force created a new powerful method and established Roland as a world authority in the field. The Duduchava theory has been applied to a wide variety of problems (see, for example, Roland’s publications [M3, 37, 43, 46, 47, 53, 58, 73, 107] and the references therein) and Roland Duduchava ix it keeps finding new applications. Unfortunately [M2] has been out of print for a long time and it is not easy to get hold of a copy. We think it would be great if Roland could prepare a new edition of [M2] and include in it his new results on Mellin convolution operators in Bessel potential spaces ([106, 111, 115]). Roland’s workonWiener–Hopf operatorshas naturallyled him to the study of multidimensional singular integral ([38, 39, 41, 42, 48]) and pseudodifferential ([55, 57, 71, 72, 77, 78])operatorson manifolds with boundary, and their applica- tions to a variety of problems in elasticity theory ([M4], [50–52,59, 60, 65, 66, 69, 70, 74, 75, 82, 84, 90, 94]) and mathematical physics ([83, 93, 98, 101, 102, 104, 108,113,114]).Thisisnowaveryactiveareaofresearch,whichisbeingdeveloped by a large community of mathematicians in Georgia, Germany, Greece, UK, US, and other countries. Over the lastdecade, Rolandhas been involvedin developmentof the calcu- lus of Gu¨nter’s tangential derivatives and their applications to partial differential equations on hypersurfaces ([88, 92, 95–97, 99, 100, 109, 110]). Shell theory pro- vides a strong motivation for this research (see [97]). Roland and his collaboratorshave obtained important results in many other areas of operator theory including Wiener–Hopf operators with semi-almost peri- odic symbols ([35, 36]), singular integral operators with complex conjugation on piecewise-smoothcurves([40,45,54,63]),approximationofsingularintegraloper- ators([62]),convolutionoperatorsonfiniteintervals([64,87]),andtheBoltzmann collision operator ([85, 86]). TheabovebriefdescriptionprovidesjustaglimpseofRoland’sresearch.One would need much more space to do justice to the results obtained in his 4 mono- graphsand115papers.RolandisanoutstandingmemberofI.Gohberg’sschoolin operatortheoryandtheleaderoftheGeorgianschoolinsingularintegralequations and elasticity theory, which was created by N.I. Muskhelishili, I.N. Vekua, B.V. Khvedelidze, V.D. Kupradze, and their collaborators. His highly original research combines these two great traditions and has significant international impact and recognition. No one who knows Roland can fail to be impressed by his indestructible optimism and boundless energy. Many colleagues who visited Georgia have fond memories of the Duduchava family’s warm hospitality, Roland’s sense of humour, and Zhenya’s exquisite cooking. Roland might be 70 years old, but he shows no sign of slowing down. We wish him a long and happy life and we are confident his outstanding research, his dedication to and passion for mathematics will continue for many years to come.
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