Around the Research of Vladimir Maz'ya I: Function Spaces

414 Pages·2010·3.2 MB·English

by  Farit Avkhadiev

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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
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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