AROUND THE RESEARCH OF VLADIMIR MAZ'YA III ANALYSIS AND APPLICATIONS INTERNATIONAL MATHEMATICAL SERIES Series Editor: Tamara Rozhkovskaya Novosibirsk, Russia 1. Nonlinear Problems in Mathematical Physics and Related Topics I. In Honor of Professor O.A. Ladyzhenskaya ² M.Sh. Birman, S. Hildebrandt, V.A. Solonnikov, N.N. Uraltseva Eds. ² 2002 2. Nonlinear Problems in Mathematical Physics and Related Topics II. In Honor of Professor O. A. Ladyzhenskaya ² M.Sh. Birman, S. Hildebrandt, V.A. Solonnikov, N.N. Uraltseva Eds. ² 2003 3. Di®erent Faces of Geometry ² S. Donaldson, Ya. Eliashberg, M. Gro- mov Eds. ² 2004 4. Mathematical Problems from Applied Logic I. Logics for the XXIst Century ² D. Gabbay, S. Goncharov, M. Zakharyaschev Eds. ² 2006 5. Mathematical Problems from Applied Logic II. Logics for the XXIst Century ² D. Gabbay, S. Goncharov, M. Zakharyaschev Eds. ² 2007 6. Instability in Models Connected with Fluid Flows. I ² C. Bardos, A. Fursikov Eds. ² 2008 7. Instability in Models Connected with Fluid Flows. II ² C. Bardos, A. Fursikov Eds. ² 2008 8. Sobolev Spaces in Mathematics I. Sobolev Type Inequalities ² V. Maz'ya Ed. ² 2009 9. Sobolev Spaces in Mathematics II. Applications in Analysis and Partial Di®erential Equations ² V. Maz'ya Ed. ² 2009 10. Sobolev Spaces in Mathematics III. Applications in Mathemat- ical Physics ² V. Isakov Ed. ² 2009 11. Around the Research of Vladimir Maz'ya I. Function Spaces ² A. Laptev Ed. ² 2010 12. Around the Research of Vladimir Maz'ya II. Partial Di®erential Equations ² A. Laptev Ed. ² 2010 13. Around the Research of Vladimir Maz'ya III. Analysis and Ap- plications ² A. Laptev Ed. ² 2010 AROUND THE RESEARCH OF VLADIMIR MAZ'YA III Analysis and Applications Editor: Ari Laptev Imperial College London, UK Royal Institute of Technology, Sweden SPRINGER TAMARA ROZHKOVSKAYA PUBLISHER Editor Prof. Ari Laptev Imperial College London Department of Mathematics London, UK Royal Institute of Technology Department of Mathematics Stockholm, Sweden This series was founded in 2002 and is a joint publication of Springer and \Tamara Rozhkovskaya Publisher." Each volume presents contribu- tions from the Volume Editors and Authors exclusively invited by the Se- ries Editor Tamara Rozhkovskaya who also prepares the Camera Ready Manuscript. This volume is distributed by \Tamara Rozhkovskaya Publisher" ([email protected]) in Russia and by Springer over the world. ISBN 978-1-4419-1344-9 (Springer) ISBN 978-5-9018-7343-4 (Tamara Rozhkovskaya Publisher) ISSN 1571-5485 Library of Congress Control Number: 0000000 °c 2010 Springer Science+Business Media, LLC All right reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Sci- ence+Business Media, LLC, 288 Springer Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identi¯ed as such, is not to be taken as an expression of options as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com Vladimir Maz'ya was born on December 31, 1937, in Leningrad (present day, St. Petersburg) in the former USSR. His ¯rst mathematical article was published in Doklady Akad. Nauk SSSR when he was a fourth-year student of the Leningrad State University. From 1961 till 1986 V. Maz'ya held a senior research fellow position at the Research Institute of Mathematics and Mechanics of LSU, and then, during 4 years, he headed the Labora- tory of Mathematical Models in Mechanics at the Institute of Engineering Studies of the Academy of Sciences of the USSR. Since 1990, V. Maz'ya lives in Sweden. At present, Vladimir Maz'ya is a Professor V. Maz'ya. 1950 Emeritus at LinkÄopingUniversity and Professor at Liverpool University. He was elected a Member of Royal Swedish Academy of Sciences in 2002. The list of publications of V. Maz'ya contains 20 books and more than 450 re- search articles covering diverse areas in Analysis and containing numerous fundamental results and fruitful techniques. Research activities of Vladimir Maz'ya have strongly influenced the development of many branches in Anal- ysis and Partial Di®erential Equations, which are clearly highlighted by the contributions to this collection of 3 volumes, where the world-recognized spe- cialists present recent advantages in the areas I. Function Spaces (Sobolev type spaces, isoperimetric and capacitary inequalities in di®erent contexts, sharp constants, traces, extnesion operators, etc.) II. Partial Di®erential Equations (asymptotic analysis, homogenization, boundary value problems, boundary integral equations, etc.) III. Analysis and Applications (various problems including the oblique derivative problem, ill-posed problems, etc. ) The monograph Theory of Sobolev Multipliers by Vladimir Maz'ya and his wife Tatyana Shaposhnikova was published in 2009 (Springer). In 2003, the French Academy of Sciences awarded V. Maz'ya and T. Shaposhnikova the Verdaguer prize for the book Jacques Hadamard, a universal mathematician. Analysis and Applications Contributions of Vladimir Maz'ya The following should be also included into the long list of influential contri- butions of Vladimir Maz'ya. | In 1989, V. Maz'ya and V. Kozlov developed new iterative procedures for solving ill-posed problems of mathematical physics. Unlike widely used numerical methods, where the equations are perturbed, they proposed to con- struct iteratively solutions to problems for the original equation and provide the convergence by an appropriate choice of the boundary conditions at each step. In 1994, V. Kozlov, V. Maz'ya, and A. Fomin established the unique- ness theorem for the inverse problem of coupled thermoelasticity. An inverse problem connected with a wave motion was also discussed in the paper by H. Hellsten, V. Maz'ya, and B. Vainberg (2003). | Explicit criteria of the Lp-contractivity of semigroups generated by el- liptic equations and systems were obtained by V. Maz'ya and A. Cialdea in 2005{2006. | In 1972, V. Maz'ya made a breakthrough in the study of the oblique derivative problem (Poincar¶eproblem). General degenerate elliptic pseudod- i®erential operators on a compact manifold were studied by V. Maz'ya and B. Paneyah (1974). | V. Maz'ya obtained signi¯cant results in mathematical elasticity and fluid mechanics. In 1967, V. Maz'ya and S. Mikhlin published a paper concerning the Cosserat spectrum associated with the Lam¶eequations of elastostatics. A method of computation of stress intensity factors in fracture mechanics was developed by V. Maz'ya and B. Plamenevskii (1974). | Several important results concerning SchrÄodingerand general second order di®erential operators were obtained by V. Maz'ya with coauthors. In particular, two papers with M. Shubin contain necessary and su±cient results on the discreteness of spectra for the SchrÄodingeroperator. | Necessary and su±cient conditions ensuring various estimates for gen- eral di®erential operators with pseudodi®erential coe±cients were found by I. Gelman and V. Maz'ya. These results were collected in their book of 1981. | Among Maz'ya's contributions to the complex function theory are sharp approximation theorems and improvement of Carleson's uniqueness theorems obtained together with V. Khavin (1968-1969). A new uni¯ed approach to Hadamard type real part theorems for analytic functions was recently pro- posed by G. Kresin and V. Maz'ya. | In the late 1980's, V. Maz'ya invented the so-called \approximate ap- proximations," which deliver a numerical approximation within the desired accuracy even though the approximation process does not necessarily con- verge as the grid size tends to zero. On this subject, the book by V. Maz'ya and G. Schmidt was published in 2007. xvi Main Topics Main Topics In this volume, the following topics are discussed: ² Obstacle type variational inequality with biharmonic di®erential operator. /Adams{Hrynkiv{Lenhart/ ² The Beurling minimum principle (studied by V. Maz'ya in 1972) is applied for treating the minimal thinness of nontangentially accessible domains. /Aikawa/ ² The Lp-dissipativity of partial di®erential operators and the Lp-contracti- vity of the generated semigroups Review of results by V. Maz'ya, G. Kresin, M. Langer, and A. Cialdea. /Cialdea/ ² Uniqueness and nonuniqueness in inverse hyperbolic problems and the existence of black (white) holes. /Eskin/ ² Global exponential bounds for Green's functions for a broad class of dif- ferential and integral equations with possibly singular coe±cients, data, and boundaries of the domains. /Frazier{Verbitsky/ ² Properties of spectral minimal partitions in the case of the sphere. /Hel®er{T.Ho®mann-Ostenhof{Terracini/ ² The boundedness of integral operators from Besov spaces on the boundary of a Lipschitz domain ­ into weighted Sobolev spaces of functions in ­ /D.Mitrea{M.Mitrea{Monniaux/ ² The Cwikel{Lieb{Rozenblum and Lieb{Thirring inequalities for operators on functions in metric spaces. Applications to spectral problems with the SchrÄodingeroperator. /Molchanov{Vainberg/ ² The Weyl formula for the Laplace operator on a domain under minimal assumptions on the boundary. Estimates for the remainder. /Netrusov{Safarov/ ² The W 2;p-theory of a degenerate oblique derivative problem for second order uniformly elliptic operators. /Palagachev/ ² The Hardy integral operator. Weighted inequalities in the integral and supremum form. /Pick/ ² Finite rank Toeplitz operators and applications. /Rozenblum/ ² Estimates for the resolvent of a non-selfadjoint pseudodi®erential operator, An approach based on estimates of an associated semigroup. /Sjostrand/ http://www.springer.com/978-1-4419-1344-9.