Concentration Compactness: Functional-Analytic Theory of Concentration Phenomena (De Gruyter Series in Nonlinear Analysis and Applications, Band 33)

231 Pages·2020·5.163 MB·English

by  Cyril Tintarev

CyrilTintarev
ConcentrationCompactness
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De Gruyter Series in Nonlinear
Analysis and Applications
|
Editor-inChief
JürgenAppell,Würzburg,Germany
Editors
CatherineBandle,Basel,Switzerland
AlainBensoussan,Richardson,Texas,USA
AvnerFriedman,Columbus,Ohio,USA
MikioKato,Tokyo,Japan
WojciechKryszewski,Torun,Poland
UmbertoMosco,Worcester,Massachusetts,USA
LouisNirenberg,NewYork,USA
SimeonReich,Haifa,Israel
AlfonsoVignoli,Rome,Italy
VicenţiuD.Rădulescu,Krakow,Poland
Volume 33
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Cyril Tintarev
Concentration
Compactness
|
Functional-Analytic Theory of Concentration Phenomena
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MathematicsSubjectClassification2010
Primary:46B50,35J60,46N20;Secondary:46E35,35A15
Author
CyrilTintarev
Technion
DepartmentofMathematics
32000Haifa
Israel
[email protected]
ISBN978-3-11-053034-6
e-ISBN(PDF)978-3-11-053243-2
e-ISBN(EPUB)978-3-11-053058-2
ISSN0941-813X
LibraryofCongressControlNumber:2019952753
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2020WalterdeGruyterGmbH,Berlin/Boston
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
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ToSonia
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Preface
The subject of this book is convergence of sequences in Banach spaces without a
given compact embedding, or more specifically, structural representation of such
sequences,knowninapplicationsasconcentrationcompactness,addressedonthe
functional-analyticlevel.
Concentrationcompactnessbecameastandardtoolofanalysisofpartialdiffer-
entialequationssincethepublicationofcelebratedpapers[83,84]byP.-L.Lions,fol-
lowedbytheprofiledecompositionapproachintroducedbyStruwe[119],generalized
togeneralsequencesinSobolevspacesbySolimini[112],andfurthergeneralizedto
sequencesinHilbertandBanachspaces,respectively,in[104]and[113].
Thisbookisasequeltoanearliermonograph[127],whosepurposewastogive
afunctional-analytictheoryofconcentrationcompactnessingeneralHilbertspaces,
and to illustrate this abstract approach by applications to calculus of variations,
mostlyinthesettingsofLions.Inthepresentbook,thefocusisshiftedfromsampling
theknownapplicationstoabroaderpresentationofthemethod,basedonthecur-
rentstateofart.ThebookextendsanalysisofconcentrationfromHilberttoBanach
spaces,andpresentsrealizationsofconcentrationcompactnessinavarietyoffunc-
tionalspaces,while[127]dealtonlywithSobolevspaces.Nowintoconsiderationcome
BesovandTriebel–Lizorkinspaces,embeddingsintospacesofcontinuousfunctions,
embeddings associated with the Moser–Trudinger inequality, Strichartz embedding
forthenonlinearSchrödingerequation,andtheaffineSobolevinequality.Thebook
alsoextendsthenotionofprofiledecompositiontofunctionalspacesthatdonothave
anontrivialgroupofinvariance.
Central to this book is the notion of cocompact embedding, which in [127] ap-
pearsonlyimplicitly.CocompactnessofanembeddingoftwoBanachspacesisaprop-
ertysimilartobutweakerthancompactness,anditplayscentralroleinhavingwell-
structuredprofiledecompositionsforboundedsequences–sumofasymptoticallyde-
coupled“blowups.”
Chapter1givesabriefintroductiontothebasicnotionsofthetheoryandexam-
ples of an “orderlyloss”of compactness(profiledecomposition)in presence of co-
compactembeddings.Chapter2containstechnicalpreliminariesconcerningDelta-
convergence, a less-known cousin of weak convergence, involved in the profile de-
compositionforBanachspaces,whichareconsideredinChapter4togetherwithits
realizationinSobolevandotherscale-invariantfunctionspaces.Chapter3sumsup
knownresultsoncocompactnessrelativetotherescalinggroup(actionsoftransla-
tions and dilations), in Besov and Triebel–Lizorkin spaces (with Sobolev and frac-
tionalSobolevspacesasaparticularcase),aswellascocompactnessofanembedding
oftheMoser–Trudinger-typerelativetoadifferentgroupoflogarithmicdilations.
Chapters 5 through 9 can be read independently one of the other. Chapter 5
presentsfurthercocompactembeddingsandprofiledecompositions.Chapter6dis-
https://doi.org/10.1515/9783110532432-201
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VIII | Preface
cussesdefectofcompactnessforsequencesrestrictedtodifferentsubspaces.Chap-
ters 7 and 8 deal with profile decompositions that do not follow from the general
framework of Chapter 4 – for nonreflexive spaces and for Sobolev spaces without
invariance. Chapter 9 presents a small selection of applications of concentration
methodstosemilinearellipticequations.
Corrections,supportingmaterials,etc.relatedtothisbook,willappearontheau-
thor’spersonalwebsite,http://sites/google.com/site/tintarev.
Thebookwaswrittenindifficultcircumstances,assince2016theauthorwassub-
jectedbyhisformeremployertoacompletetravelban(includinghost-andself-funded
travel),togetherwithfurtherrestrictions,whichbroughttheauthortoleavehisjob
atUppsalaUniversity.TheauthorexpresseshiswarmgratitudetoAcademicRights
Watch and his colleagues and collaborators at Technion, University of Toulouse –
LaCapitole,TataInstituteforfundamentalresearch,UniversityofBariandPolitec-
nicUniversityofBari,fortheirunwaveringsupportofhisacademicrights.Hethanks
TorbjörnOhlsson,attorneyatlaw,whonegotiatedauthor’scontinuedaccesstothe
libraryresourcesofhisformeremployer.
Theworkonthisbookwascompletedduringtheauthor’sstayasLadyDavisVis-
itingProfessoratTechnion–IsraelInstituteofTechnology.
Haifa,December2019
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Contents
Preface|VII
1 Profiledecomposition:astructureddefectofcompactness|1
1.1 Cocompactembeddings:definitionandexamples|1
1.2 Profiledecomposition|5
1.3 Brezis–Lieblemma|7
1.4 Lions’lemmafortheMoser–Trudingerfunctional|11
1.5 Bibliographicnotes|13
2 Delta-convergenceandweakconvergence|15
2.1 DefinitionofDelta-convergence|15
2.2 Chebyshevandasymptoticcenters.Delta-completenessand
Delta-compactness|16
2.3 Rotundmetricspaces|18
2.4 OpialconditionandVanDulstnorm|21
2.5 Defectofenergy.Brezis–LieblemmawithDelta-convergence|25
2.6 Bibliographicnotes|27
3 Cocompactembeddingswiththerescalinggroup|29
3.1 Definitionsandelementarypropertiesofcocompactness|29
3.2 CocompactnessofthelimitingSobolevembedding|30
3.3 EmbeddingḢ1,p(ℝN)󳨅→Lp∗,p(ℝN)isnotcocompact|34
3.4 Cocompactnessandexistenceofminimizers|35
3.5 CocompactembeddingsofBesovandTriebel–Lizorkinspaces|37
3.6 Cocompactnessandinterpolation|40
3.7 CocompactembeddingsofinhomogeneousBesovspaces|44
3.8 CocompactembeddingsofintersectionswithLp(ℝN)|49
3.9 Cocompactnessoftraceembeddings|51
3.10 Spacescocompactlyembeddedintothemselves|54
3.11 CocompactnessoftheradialMoser–Trudingerembedding|55
3.12 Bibliographicnotes|57
4 ProfiledecompositioninBanachspaces|59
4.1 Profiledecomposition|59
4.2 OpialconditioninBesovandTriebel–Lizorkinspaces|61
4.3 ProofofTheorem4.1.6|63
4.4 Profiledecompositioninthedualspace|66
4.5 Profiledecompositionforvector-valuedfunctions|68
4.6 ProfiledecompositioninBesov,Triebel–Lizorkin,andSobolev
spaces|70
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