SOBOLEV SPACES IN MATHEMATICS III APPLICATIONS IN MATHEMATICAL PHYSICS INTERNATIONAL MATHEMATICAL SERIES Series Editor:Tamara Rozhkovskaya Novosibirsk, Russia 1. NonlinearProblemsinMathematicalPhysicsandRelatedTopics I. In Honor of Professor O.A. Ladyzhenskaya • M.Sh. Birman, S. Hildebrandt, V.A. Solonnikov, N.N. Uraltseva Eds. • 2002 2. NonlinearProblemsinMathematicalPhysicsandRelatedTopics II. In Honor of Professor O.A. Ladyzhenskaya • M.Sh. Birman, S. Hildebrandt, V.A. Solonnikov, N.N. Uraltseva Eds. • 2003 3. Different Faces of Geometry • S. Donaldson,Ya. Eliashberg,M. Gro- mov Eds. • 2004 4. Mathematical Problems from Applied Logic I. Logics for the XXIst Century • D. Gabbay, S. Goncharov, M. Zakharyaschev Eds. • 2006 5. Mathematical Problems from Applied Logic II. Logics for the XXIst Century • D. Gabbay, S. Goncharov, M. Zakharyaschev Eds. • 2007 6. Instability in Models Connected with Fluid Flows. I • C. Bardos, A. Fursikov Eds. • 2008 7. Instability in Models Connected with Fluid Flows. II•C.Bardos, A. Fursikov Eds. • 2008 8. Sobolev Spaces in Mathematics I. Sobolev Type Inequalities • V. Maz’ya Ed. • 2009 9. Sobolev Spaces in Mathematics II. Applications in Analysis and Partial Differential Equations • V. Maz’ya Ed. • 2009 10.Sobolev Spaces in Mathematics III. Applications in Mathemat- ical Physics • V. Isakov Ed. • 2009 SOBOLEV SPACES IN MATHEMATICS III Applications in Mathematical Physics Victor Isakov Editor: Wichita State University, USA 123 Tamara Rozhkovskaya Publisher Editor Prof. Victor Isakov Department of Mathematics and Statistics Wichita State University Wichita KS 67260-0033 USA 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 andAuthorsexclusivelyinvitedbytheSeriesEditorTamaraRozhkovskaya whoalsopre- parestheCameraReadyManuscript.Thisvolumeisdistributedby“TamaraRozhkovskaya Publisher”([email protected]) inRussiaandbySpringeroveralltheworld. ISBN 978-0-387-85651-3 e-ISBN 978-0-387-85652-0 ISBN 978-5-901873-28-1 (Tamara RozhkovskayaPublisher) ISSN1571-5485 LibraryofCongressControlNumber:2008937487 (cid:2)c 2009SpringerScience+Business Media,LLC Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithout thewrittenpermissionofthepublisher(SpringerScience+BusinessMedia,LLC,233Spring Street, New York, NY 10013, USA), except for brief excerpts inconnection with reviews orscholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now knownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms, eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionasto whether ornottheyaresubjecttoproprietaryrights. Printedonacid-freepaper. 9 8 7 6 5 4 3 2 1 springer.com To the memory of Sergey L’vovichSobolev on the occasion of his centenary Main Topics Sobolev’s discoveries of the 1930’s have a strong influence on de- velopment of the theory of partial differential equations, analysis, mathematicalphysics,differentialgeometry,andotherfieldsofmath- ematics. The three-volume collection Sobolev Spaces in Mathematics presents the latest results in the theory of Sobolev spaces and appli- cations from leading experts in these areas. I. Sobolev Type Inequalities In 1938, exactly 70 years ago, the original Sobolev inequality (an embed- ding theorem) was published in the celebrated paper by S.L. Sobolev “On a theorem of functional analysis.” By now, the Sobolev inequality and its numerous versions continue to attract attention of researchers because of the central role played by such inequalities in the theory of partial differ- ential equations, mathematical physics, and many various areas of analysis and differential geometry. The volume presents the recent study of different Sobolev type inequalities, in particular, inequalities on manifolds, Carnot– Carath´eodoryspaces,andmetric measurespaces,traceinequalities, inequal- itieswithweights,thesharpnessofconstantsininequalities,embeddingtheo- remsindomainswithirregularboundaries,thebehaviorofmaximalfunctions in Sobolev spaces, etc. Some unfamiliar settings of Sobolev type inequalities (forexample,ongraphs)arealsodiscussed.Thevolumeopenswiththesurvey article “My Love Affair with the Sobolev Inequality” by David R. Adams. II. Applications in Analysis and Partial Differential Equations Sobolev spaces become the established language of the theory of partial dif- ferential equations and analysis. Among a huge variety of problems where Sobolevspacesareused,thefollowingimportanttopicsareinthefocusofthis volume:boundaryvalueproblemsindomainswithsingularities,higherorder partialdifferentialequations,nonlinearevolutionequations,localpolynomial approximations,regularityfor the Poissonequationin cones,harmonic func- tions,inequalitiesinSobolev–Lorentzspaces,propertiesoffunctionspacesin cellular domains, the spectrum of a Schro¨dinger operator with negative po- tential,thespectrumofboundaryvalueproblemsindomainswithcylindrical and quasicylindricaloutlets to infinity, criteria for the complete integrability of systems of differential equations with applications to differential geome- try,someaspectsofdifferentialformsonRiemannianmanifoldsrelatedtothe Sobolevinequality,aBrownianmotiononaCartan–Hadamardmanifold,etc. Two short biographical articles with unique archive photos of S.L. Sobolev are also included. viii MainTopics III. Applications in Mathematical Physics ThemathematicalworksofS.L.Sobolevwerestronglymotivatedbyparticu- larproblemscomingfromapplications.Theapproachandideasofhisfamous book“ApplicationsofFunctionalAnalysisinMathematicalPhysics”of1950 turned out to be very influential and are widely used in the study of various problemsofmathematicalphysics.The topicsofthis volumeconcernmathe- maticalproblems,mainly from controltheory andinverseproblems,describ- ing various processes in physics and mechanics, in particular, the stochastic Ginzburg–Landaumodel with white noise simulating the phenomenonofsu- perconductivity in materials under low temperatures, spectral asymptotics for the magnetic Schro¨dinger operator, the theory of boundary controllabil- ity for models of Kirchhoff plate and the Euler–Bernoulli plate with various physically meaningful boundary controls, asymptotics for boundary value problems in perforated domains and bodies with different type defects, the Finsler metric inconnectionwiththe study ofwavepropagation,the electric impedance tomography problem, the dynamical Lam´e system with residual stress, etc. Contents I. Sobolev Type Inequalities Vladimir Maz’ya Ed. My Love Affair with the Sobolev Inequality ..............................1 David R. Adams Maximal Functions in Sobolev Spaces ...................................25 Daniel Aalto and Juha Kinnunen Hardy Type Inequalities Via Riccati and Sturm–Liouville Equations ....69 Sergey Bobkov and Friedrich G¨otze Quantitative Sobolev and Hardy Inequalities, and Related Symmetrization Principles ..............................................87 Andrea Cianchi Inequalities of Hardy–Sobolev Type in Carnot–Carath´eodorySpaces ...117 Donatella Danielli, Nicola Garofalo, and Nguyen Cong Phuc Sobolev Embeddings and Hardy Operators ............................153 David E. Edmunds and W. Desmond Evans Sobolev Mappings between Manifolds and Metric Spaces ...............185 Piotr Haj(cid:3)lasz A Collection of Sharp Dilation Invariant Integral Inequalities for Differentiable Functions ............................................223 Vladimir Maz’ya and Tatyana Shaposhnikova Optimality of Function Spaces in Sobolev Embeddings .................249 Luboˇs Pick On the Hardy–Sobolev–Maz’yaInequality and Its Generalizations .....281 Yehuda Pinchover and Kyril Tintarev Sobolev Inequalities in Familiar and Unfamiliar Settings ...............299 Laurent Saloff-Coste A Universality Property of Sobolev Spaces in Metric Measure Spaces ..345 Nageswari Shanmugalingam Cocompact Imbeddings and Structure of Weakly Convergent Sequences .............................................................361 Kiril Tintarev Index..................................................................377 x SobolevSpaces inMathematics I–III II. Applications in Analysis and Partial Differential Equations Vladimir Maz’ya Ed. On the Mathematical Works of S.L. Sobolev in the 1930s ................1 Vasilii Babich Sobolev in Siberia ......................................................11 Yuri Reshetnyak Boundary Harnack Principle and the Quasihyperbolic Boundary Condition ..............................................................19 Hiroaki Aikawa Sobolev Spaces and their Relatives: Local Polynomial Approximation Approach ...............................................31 Yuri Brudnyi Spectral Stability of Higher Order Uniformly Elliptic Operators .........69 Victor Burenkov and Pier Domenico Lamberti Conductor Inequalities and Criteria for Sobolev-Lorentz Two-Weight Inequalities ...............................................103 Serban Costea and Vladimir Maz’ya Besov Regularity for the PoissonEquation in Smooth and Polyhedral Cones ......................................................123 Stephan Dahlke and Winfried Sickel Variational Approach to Complicated Similarity Solutions of Higher Order Nonlinear Evolution PartialDifferential Equations .......147 Victor Galaktionov, Enzo Mitidieri, and Stanislav Pokhozhaev L -Cohomology of Riemannian Manifolds with Negative Curvature ...199 q,p Vladimir Gol’dshtein and Marc Troyanov Volume Growth and Escape Rate of Brownian Motion on a Cartan–HadamardManifold .........................................209 Alexander Grigor’yan and Elton Hsu Sobolev Estimates for the Green Potential Associated with the Robin–Laplacian in Lipschitz Domains Satisfying a Uniform Exterior Ball Condition .....................................227 Tu¨nde Jakab, Irina Mitrea, and Marius Mitrea Properties of Spectra of Boundary Value Problems in Cylindrical and Quasicylindrical Domains ...........................261 Sergey Nazarov Estimates for Completeley Integrable Systems of Differential Operators and Applications ...........................................311 Yuri Reshetnyak Contents xi Counting Schro¨dinger Boundstates: Semiclassics and Beyond ..........329 Grigori Rozenblum and Michael Solomyak Function Spaces on Cellular Domains ..................................355 Hans Triebel Index..................................................................387 III. Applications in Mathematical Physics Victor Isakov Ed. Preface ..................................................................1 Victor Isakov Geometrization of Rings as a Method for Solving Inverse Problems .......5 Mikhail Belishev The Ginzburg–Landau Equations for Superconductivity with Random Fluctuations ...................................................25 Andrei Fursikov, Max Gunzburger, and Janet Peterson Carleman Estimates with Second Large Parameter for Second Order Operators ......................................................135 Victor Isakov and Nanhee Kim Sharp Spectral Asymptotics for Dirac Energy ..........................161 Victor Ivrii Linear Hyperbolic and PetrowskiType PDEs with Continuous Boundary Control → Boundary Observation Open Loop Map: Implication on Nonlinear Boundary Stabilization with Optimal Decay Rates ..................................................187 Irena Lasiecka and Roberto Triggiani Uniform Asymptotics of Green’s Kernels for Mixed and Neumann Problems in Domains with Small Holes and Inclusions .................277 Vladimir Maz’ya and Alexander Movchan Finsler Structures and Wave Propagation ..............................317 Michael Taylor Index..................................................................335