Recent Trends in Operator Theory and Partial Differential Equations The Roland Duduchava Anniversary Volume

313 Pages·2017·2.483 MB·English

by  Maz'ya, Vladimir, Natroshvili, David, Shargorodsky, Eugene, Wendland, Wolfgang L et al. (eds.)

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

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Contents
V. Maz’ya, D. Natroshvili, E. Shargorodsky and W.L. Wendland
Roland Duduchava .................................................. vii
A. B¨ottcher
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.