Around the Research of Vladimir Maz'ya I: Function Spaces PDF
Preview Around the Research of Vladimir Maz'ya I: Function Spaces
INTERNATIONAL MATHEMATICAL SERIES Series Editor:Tamara Rozhkovskaya Novosibirsk, Russia This series was founded in 2002 and is a joint publication of Springer and “Tamara Rozhkovskaya Publisher.” Each volume presents contributions from the Volume Editors and Authors exclusively invited by the Series 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 all the world. Forothertitles published in this series, go to www.springer.com/series/6117 AROUND THE RESEARCH OF VLADIMIR MAZ’YA I Function Spaces Ari Laptev Editor: Imperial College London, UK Royal Institute of Technology, Sweden SPRINGER TAMARAROZHKOVSKAYAPUBLISHER Editor Ari Laptev Department of Mathematics Imperial College London Huxley Building, 180 Queen’s Gate London SW7 2AZ United Kingdom [email protected] ISSN 1571-5485 e-ISSN 1574-8944 ISBN 978-1-4419-1340-1 e-ISBN 978-1-4419-1341-8 ISBN 978-5-9018-7341-0 (Tamara Rozhkovskaya Publisher) DOI 10.1007/978-1-4419-1341-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009941304 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media LLC, 233 Spring 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 identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Vladimir Maz’ya was born on Decem- ber 31, 1937, in Leningrad (present day, St. Petersburg) in the former USSR. His first 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, dur- ing 4 years, he headed the Laboratory 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 Emeritus at Link¨oping University and Professor at Liv- SergeyL.Sobolev(left)and erpool University. He was elected a Member VladimirG.Maz’ya(right). ofRoyalSwedishAcademyofSciencesin2002. Novosibirsk,1978 The list of publications of V. Maz’ya contains 20 books and more than 450 research 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 Differential Equations, which are clearly highlighted by the contributionstothiscollectionof3volumes,wheretheworld-recognizedspe- cialists present recent advantages in the following areas: I.FunctionSpaces.VariousaspectsofthetheoryofSobolevspaces,isoperi- metricandcapacitaryinequalities,Hardy,Sobolev,andPoincar´einequalities in different contexts, sharp constants, extension operators, traces, weighted Sobolev spaces, Orlicz–Sobolev spaces, Besov spaces, etc. II. Partial Differential Equa- tions. Asymptotic analysis, multi- scaleasymptoticexpansions,homog- enization, boundary value problems indomainswithsingularities,bound- aryintegralequations,mathematical theory of water waves, Wiener regu- larity of boundary points, etc. III. Analysis and Applications. Various problems including the oblique derivative problem, spec- tral properties of the Schr¨odinger LaurentSchwartz(left)and VladimirMaz’ya(right).Paris,1992 operator, ill-posed problems, etc. vi Contents I. Function Spaces Ari Laptev Ed. Hardy Inequalities for Nonconvex Domains ..............................1 Farit Avkhadiev and Ari Laptev Distributions with Slow Tails and Ergodicity of Markov Semigroups in Infinite Dimensions ..................................................13 Sergey Bobkov and Boguslaw Zegarlinski On Some Aspects of the Theory of Orlicz–Sobolev Spaces ...............81 Andrea Cianchi Mellin Analysis of Weighted Sobolev Spaces with Nonhomogeneous Norms on Cones .......................................................105 Martin Costabel, Monique Dauge, and Serge Nicaise Optimal Hardy–Sobolev–Maz’ya Inequalities with Multiple Interior Singularities ...................................................137 Stathis Filippas, Achilles Tertikas, and Jesper Tidblom Sharp Fractional Hardy Inequalities in Half-Spaces ....................161 Rupert L. Frank and Robert Seiringer Collapsing Riemannian Metrics to Sub-Riemannian and the Geometry of Hypersurfaces in Carnot Groups ......................169 Nicola Garofalo and Christina Selby Sobolev Homeomorphisms and Composition Operators ................207 Vladimir Gol’dshtein and Aleksandr Ukhlov Extended Lp Dirichlet Spaces .........................................221 Niels Jacob and Ren´e L. Schilling Characterizations for the Hardy Inequality ............................239 Juha Kinnunen and Riikka Korte Geometric Properties of Planar BV-Extension Domains ...............255 PekkaKoskela,MicheleMirandaJr.,andNageswariShanmugalingam On a New Characterization of Besov Spaces with Negative Exponents .273 Moshe Marcus and Laurent V´eron Isoperimetric Hardy Type and Poincar´e Inequalities on Metric Spaces ..285 Joaquim Mart´ın and Mario Milman Gauge Functions and Sobolev Inequalities on Fluctuating Domains ....299 Eric Mbakop and Umberto Mosco A Converse to the Maz’ya Inequality for Capacities under Curvature Lower Bound ...............................................321 Emanuel Milman Pseudo-Poincar´e Inequalities and Applications to Sobolev Inequalities .349 Laurent Saloff-Coste The p-Faber-Krahn Inequality Noted ..................................373 Jie Xiao Index..................................................................391 References to Maz’ya’s Publications Made in Volume I .................393 Contents vii II. Partial Differential Equations Ari Laptev Ed. Large Solutions to Semilinear Elliptic Equations with Hardy Potential and Exponential Nonlinearity ..................................1 Catherine Bandle, Vitaly Moroz, and Wolfgang Reichel Stability Estimates for Resolvents, Eigenvalues, and Eigenfunctions of Elliptic Operators on Variable Domains ..............................23 GerassimosBarbatis,VictorI.Burenkov,andPierDomenicoLamberti Operator Pencil in a Domain with Concentrated Masses. A Scalar Analog of Linear Hydrodynamics ..............................61 Gregory Chechkin Selfsimilar Perturbation near a Corner: Matching Versus Multiscale Expansions for a Model Problem .......................................95 Monique Dauge, S´ebastien Tordeux, and Gr´egory Vial Stationary Navier–Stokes Equation on Lipschitz Domains in Riemannian Manifolds with Nonvanishing Boundary Conditions .......135 Martin Dindoˇs On the Regularity of Nonlinear Subelliptic Equations ..................145 Andr´as Domokos and Juan J. Manfredi Rigorous and Heuristic Treatment of Sensitive Singular Perturbations Arising in Elliptic Shells ...............................................159 Yuri V. Egorov, Nicolas Meunier, and Evariste Sanchez-Palencia On the Existence of Positive Solutions of Semilinear Elliptic Inequalities on Riemannian Manifolds .................................203 Alexander Grigor’yan and Vladimir A. Kondratiev Recurrence Relations for Orthogonal Polynomials and Algebraicity of Solutions of the Dirichlet Problem ..................................219 Dmitry Khavinson and Nikos Stylianopoulos On First Neumann Eigenvalue Bounds for Conformal Metrics ..........229 Gerasim Kokarev and Nikolai Nadirashvili Necessary Condition for the Regularity of a Boundary Point for Porous Medium Equations with Coefficients of Kato Class ..........239 Vitali Liskevich and Igor I. Skrypnik The Problem of Steady Flow over a Two-Dimensional Bottom Obstacle ..............................................................253 Oleg Motygin and Nikolay Kuznetsov Well Posedness and Asymptotic Expansion of Solution of Stokes Equation Set in a Thin Cylindrical Elastic Tube .......................275 Grigory P. Panasenko and Ruxandra Stavre On Solvability of Integral Equations for Harmonic Single Layer Potential on the Boundary of a Domain with Cusp ....................303 Sergei V. Poborchi H¨older Estimates for Green’s Matrix of the Stokes System in Convex Polyhedra .....................................................315 Ju¨rgen Roßmann viii Contents Boundary Integral Methods for Periodic Scattering Problems ..........337 Gunther Schmidt Boundary Coerciveness and the Neumann Problem for 4th Order Linear Partial Differential Operators ...................................365 Gregory C. Verchota Index .................................................................379 References to Maz’ya’s Publications Made in Volume II ................381 III. Analysis and Applications Ari Laptev Ed. Optimal Control of a Biharmonic Obstacle Problem ......................1 David R. Adams, Volodymyr Hrynkiv, and Suzanne Lenhart Minimal Thinness and the Beurling Minimum Principle .................25 Hiroaki Aikawa Progress in the Problem of the Lp-Contractivity of Semigroups for Partial Differential Operators ...........................................47 Alberto Cialdea Uniqueness and Nonuniqueness in Inverse Hyperbolic Problems and the Black Hole Phenomenon ............................................77 Gregory Eskin Global Green’s Function Estimates ....................................105 Michael W. Frazier and Igor E. Verbitsky On Spectral Minimal Partitions: the Case of the Sphere ...............153 BernardHelffer,ThomasHoffmann-Ostenhof,andSusannaTerracini Weighted Sobolev Space Estimates for a Class of Singular Integral Operators .............................................................179 Dorina Mitrea, Marius Mitrea, and Sylvie Monniaux On general Cwikel–Lieb–Rozenblum and Lieb–Thirring Inequalities ....201 Stanislav Molchanov and Boris Vainberg Estimates for the Counting Function of the Laplace Operator on Domains with Rough Boundaries ...................................247 Yuri Netrusov and Yuri Safarov W2,p-Theory of the Poincar´e Problem .................................259 Dian K. Palagachev Weighted Inequalities for Integral and Supremum Operators ...........279 Luboˇs Pick Finite Rank Toeplitz Operators in the Bergman Space .................331 Grigori Rozenblum Resolvent Estimates for Non-Selfadjoint Operators via Semigroups .....359 Johannes Sj¨ostrand Index .................................................................385 References to Maz’ya’s Publications Made in Volume III ...............387 Contributors Editor Ari Laptev President TheEuropeanMathematicalSociety Professor HeadofDepartment DepartmentofMathematics ImperialCollegeLondon HuxleyBuilding,180Queen’sGate LondonSW72AZ,UK [email protected] Professor DepartmentofMathematics RoyalInstituteofTechnology 10044Stockholm,Sweden [email protected] Ari Laptev is a world-recognized specialist in Spectral Theory of Differential Operators.Hediscoveredanumberofsharpspectralandfunctionalinequali- ties.Inparticular,jointlywithhisformerstudentT.Weidl,A.Laptevproved sharp Lieb–Thirring inequalities for the negative spectrum of multidimen- sional Schr¨odinger operators, a problem that was open for more than twenty five years. A. Laptev was brought up in Leningrad (Russia). In 1971, he graduated from the Leningrad State University and was appointed as a researcher and then as an Assistant Professor at the Mathematics and Mechanics Depart- ment of LSU. In 1982, he was dismissed from his position at LSU due to his marriage to a British subject. Only after his emigration from the USSR in 1987 he was able to continue his career as a mathematician. Then A. Laptev wasemployedinSweden,firstasalectureratLink¨opingUniversityandthen from1992attheRoyalInstituteofTechnology(KTH).In1999,hebecamea professoratKTHandalsoViceChairmanofitsDepartmentofMathematics. From January 2007 he is employed by Imperial College London where from September 2008 he is the Head of Department of Mathematics. A. Laptev was the Chairman of the Steering Committee of the five years long ESF Programme SPECT, the President of the Swedish Mathematical Society from 2001 to 2003, and the President of the Organizing Committee of the Fourth European Congress of Mathematics in Stockholm in 2004. He is now the President of the European Mathematical Society for the period January 2007–December 2010. Authors David R. Adams Vol. III Andrea Cianchi Vol. I UniversityofKentucky Universit`adiFirenze Lexington,KY40506-0027 PiazzaGhiberti27,50122Firenze USA ITALY [email protected] cianchi@unifi.it Hiroaki Aikawa Vol. III Martin Costabel Vol. I HokkaidoUniversity Universit´edeRennes1 Sapporo060-0810 CampusdeBeaulieu JAPAN 35042Rennes [email protected] FRANCE [email protected] Farit Avkhadiev Vol. I Monique Dauge Vols. I, II KazanStateUniversity 420008Kazan Universit´edeRennes1 RUSSIA CampusdeBeaulieu [email protected] 35042Rennes FRANCE Catherine Bandle Vol. II [email protected] MathematischesInstitut Martin Dindoˇs Vol. II Universit¨atBasel Rheinsprung21,CH-4051Basel MaxwellInstituteof SWITZERLAND MathematicsSciences [email protected] UniversityofEdinburgh JCMBKing’sbuildingsMayfieldRd Gerassimos Barbatis Vol. II EdinbughEH93JZ UniversityofAthens UK 15784Athens [email protected] GREECE Andr´as Domokos Vol. II [email protected] CaliforniaState Sergey Bobkov Vol. I UniversitySacramento UniversityofMinnesota Sacramento95819 Minneapolis,MN55455 USA USA [email protected] [email protected] Yuri V. Egorov Vol. II Victor I. Burenkov Vol. II Universit´ePaulSabatier Universit`adegliStudidiPadova 118routedeNarbonne 63ViaTrieste,35121Padova 31062ToulouseCedex9 ITALY FRANCE [email protected] [email protected] Grigori Chechkin Vol. II Gregory Eskin Vol. III LomonosovMoscowStateUniversity UniversityofCalifornia Vorob’evyGory,Moscow LosAngeles,CA90095-1555 RUSSIA USA [email protected] [email protected] Alberto Cialdea Vol. III Nicola Garofalo Vol. I Universit`adellaBasilicata PurdueUniversity Vialedell’AteneoLucano10, WestLafayette,IN47906 85100,Potenza USA ITALY [email protected] [email protected] and
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