Isolated Singularities in Partial Differential Inequalities

364 Pages·2016·1.613 MB·English

by  Marius Ghergu, Steven D. Taliaferro

ISOLATED SINGULARITIES IN PARTIAL
DIFFERENTIAL INEQUALITIES
Inthismonographtheauthorspresentsomepowerfulmethodsfordealingwith
singularitiesinellipticandparabolicpartialdifferentialinequalities.Here,the
authorstaketheuniqueapproachofinvestigatingdifferentialinequalitiesratherthan
equations,thereasonbeingthatthesimplestwaytostudyanequationisoftento
studyacorrespondinginequality;forexample,usingsub-andsuperharmonic
functionstostudyharmonicfunctions.Anotherunusualfeatureofthepresentbook
isthatitisbasedonintegralrepresentationformulaeandnonlinearpotentials,which
havenotbeenwidelyinvestigatedsofar.Thisapproachcanalsobeusedtotackle
higher-orderdifferentialequations.
Thebookwillappealtograduatestudentsinterestedinanalysis,researchersin
pureandappliedmathematics,andengineerswhoworkwithpartialdifferential
equations.Readerswillrequireonlyabasicknowledgeoffunctionalanalysis,
measuretheory,andSobolevspaces.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationin
mathematicsormathematicalscienceandforwhichadetaileddevelopmentofthe
abstracttheoryislessimportantthanathoroughandconcreteexplorationofthe
implicationsandapplications.
BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir
subjectscomprehensively.Lessimportantresultsmaybesummarizedasexercises
attheendsofchapters.Fortechnicalities,readerscanbereferredtothe
bibliography,whichisexpectedtobecomprehensive.Asaresult,volumesare
encyclopedicreferencesormanageableguidestomajorsubjects.
Encyclopedia of Mathematics and its Applications
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
www.cambridge.org/mathematics.
114 J.Beck,CombinatorialGames
115 L.BarreiraandY.Pesin,NonuniformHyperbolicity
116 D.Z.ArovandH.Dym,J-ContractiveMatrixValuedFunctionsandRelatedTopics
117 R.Glowinski,J.-L.Lions,andJ.He,ExactandApproximateControllabilityforDistributedParameterSystems
118 A.A.BorovkovandK.A.Borovkov,AsymptoticAnalysisofRandomWalks
119 M.DezaandM.DutourSikiric´,GeometryofChemicalGraphs
120 T.Nishiura,AbsoluteMeasurableSpaces
121 M.Prest,Purity,SpectraandLocalisation
122 S.Khrushchev,OrthogonalPolynomialsandContinuedFractions
123 H.NagamochiandT.Ibaraki,AlgorithmicAspectsofGraphConnectivity
124 F.W.King,HilbertTransformsI
125 F.W.King,HilbertTransformsII
126 O.CalinandD.-C.Chang,Sub-RiemannianGeometry
127 M.Grabischetal.,AggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.Tucker,TopicsinTopologicalGraphTheory
129 J.Berstel,D.Perrin,andC.Reutenauer,CodesandAutomata
130 T.G.Faticoni,ModulesoverEndomorphismRings
131 H.Morimoto,StochasticControlandMathematicalModeling
132 G.Schmidt,RelationalMathematics
133 P.KornerupandD.W.Matula,FinitePrecisionNumberSystemsandArithmetic
134 Y.CramaandP.L.Hammer(eds.),BooleanModelsandMethodsinMathematics,ComputerScience,and
Engineering
135 V.Berthe´andM.Rigo(eds.),Combinatorics,AutomataandNumberTheory
136 A.Krista´ly,V.D.Ra˘dulescu,andC.Varga,VariationalPrinciplesinMathematicalPhysics,Geometry,and
Economics
137 J.BerstelandC.Reutenauer,NoncommutativeRationalSerieswithApplications
138 B.CourcelleandJ.Engelfriet,GraphStructureandMonadicSecond-OrderLogic
139 M.Fiedler,MatricesandGraphsinGeometry
140 N.Vakil,RealAnalysisthroughModernInfinitesimals
141 R.B.Paris,HadamardExpansionsandHyperasymptoticEvaluation
142 Y.CramaandP.L.Hammer,BooleanFunctions
143 A.Arapostathis,V.S.Borkar,andM.K.Ghosh,ErgodicControlofDiffusionProcesses
144 N.Caspard,B.Leclerc,andB.Monjardet,FiniteOrderedSets
145 D.Z.ArovandH.Dym,BitangentialDirectandInverseProblemsforSystemsofIntegralandDifferential
Equations
146 G.Dassios,EllipsoidalHarmonics
147 L.W.BeinekeandR.J.Wilson(eds.)withO.R.Oellermann,TopicsinStructuralGraphTheory
148 L.Berlyand,A.G.Kolpakov,andA.Novikov,IntroductiontotheNetworkApproximationMethodfor
MaterialsModeling
149 M.BaakeandU.Grimm,AperiodicOrderI:AMathematicalInvitation
150 J.Borweinetal.,LatticeSumsThenandNow
151 R.Schneider,ConvexBodies:TheBrunn–MinkowskiTheory(SecondEdition)
152 G.DaPratoandJ.Zabczyk,StochasticEquationsinInfiniteDimensions(SecondEdition)
153 D.Hofmann,G.J.Seal,andW.Tholen(eds.),MonoidalTopology
154 M.CabreraGarc´ıaandA´.Rodr´ıguezPalacios,Non-AssociativeNormedAlgebrasI:TheVidav–Palmerand
Gelfand–NaimarkTheorems
155 C.F.DunklandY.Xu,OrthogonalPolynomialsofSeveralVariables(SecondEdition)
156 L.W.BeinekeandR.J.Wilson(eds.)withB.Toft,TopicsinChromaticGraphTheory
157 T.Mora,SolvingPolynomialEquationSystemsIII:AlgebraicSolving
158 T.Mora,SolvingPolynomialEquationSystemsIV:BuchbergerTheoryandBeyond
159 V.Berthe´andM.Rigo(eds.),Combinatorics,WordsandSymbolicDynamics
160 B.Rubin,IntroductiontoRadonTransforms:WithElementsofFractionalCalculusandHarmonicAnalysis
161 M.GherguandS.D.Taliaferro,IsolatedSingularitiesinPartialDifferentialInequalities
162 G.MolicaBisci,V.D.Radulescu,andR.Servadei,VariationalMethodsforNonlocalFractionalProblems
163 G.TomkowiczandS.Wagon,TheBanach–TarskiParadox(SecondEdition)
Encyclopedia of Mathematics and its Applications
Isolated Singularities in Partial
Differential Inequalities
MARIUS GHERGU
UniversityCollegeDublin
STEVEN D. TALIAFERRO
TexasA&MUniversity
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107138384
©MariusGherguandStevenD.Taliaferro2016
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2016
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-1-107-13838-4Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Contents
Preface pageix
1 Representationformulaeforsingularsolutionsofpolyharmonic
andparabolicinequalities 1
1.1 Introduction 1
1.2 Harmonicinequalitiesinthepuncturedball 1
1.3 Polyharmonicinequalitiesinthepuncturedball 5
1.4 Parabolicinequalities 16
1.5 Comments 20
2 Isolatedsingularitiesofnonlinearsecond-orderelliptic
inequalities 22
2.1 Introduction 22
2.2 Casen≥3 22
2.3 Casen=2 48
2.4 Comments 63
3 Moreonisolatedsingularitiesforsemilinearellipticinequalities 65
3.1 Introduction 65
3.2 Localbehaviorofsolutions 65
3.3 Globalexistenceofsolutions 76
3.4 Casep = n+2 81
n−2
3.5 Comments 87
4 EllipticinequalitiesfortheLaplaceoperator
withHardypotential 89
4.1 Introduction 89
4.2 SomecomparisonprinciplesfortheLaplaceoperatorwith
L1potential 89
4.3 EllipticinequalitieswithHardytermsincone-likedomains 93
v
vi Contents
4.4 EllipticinequalitieswithHardytermsinsmoothdomainswith
boundarysingularities 101
4.5 EllipticinequalitieswithHardytermsandhigherdimensional
singularity 111
4.6 Comments 120
5 Singularsolutionsforsecond-ordernondivergencetype
ellipticinequalities 122
5.1 Introduction 122
5.2 Effectivedimensionandotherimportantquantities 123
5.3 AnEmden–Fowlerequation 125
5.4 Firstresult.Estimatesonthecriticalexponent 131
5.5 Secondresult.Thecriticalcase 136
5.6 Examples 138
5.7 Comments 143
6 Isolatedsingularitiesofpolyharmonicinequalities 144
6.1 Introduction 144
6.2 Polyharmonicinequalitiesinpuncturedballs 144
6.3 Polyharmonicinequalitiesinexteriordomains 153
6.4 Comments 155
7 Nonlinearbiharmonicinequalities 158
7.1 Introduction 158
7.2 Mainresults 160
7.3 Preliminaryresults 161
7.4 Casen≥3 168
7.5 Casen=2 193
7.6 Comments 199
8 Initialblow-upfornonlinearparabolicinequalities 200
8.1 Introduction 200
8.2 Optimalexponentsforaprioribounds 201
8.3 Parabolicinequalitieswithboundaryconditions 214
8.4 Comments 234
9 Semilinearellipticsystemsofdifferentialinequalities 235
9.1 Introduction 235
9.2 Casen=2 236
9.3 Casen≥3 245
9.4 Comments 259
10 Isolatedsingularitiesfornonlocalellipticsystems 264
10.1 Introduction:Whatisanonlocalequation? 264
10.2 Anonlocalellipticsystemwithconvolutionterms 265
10.3 Someintegralestimates 266
Contents vii
10.4 Casemin{α,β}≤2 270
10.5 Someauxiliaryresultsforintegralsystems 274
10.6 Caseα,β ∈(2,n) 285
10.7 Comments 292
11 Isolatedsingularitiesforsystemsofparabolicinequalities 294
11.1 Introduction 294
11.2 Preliminarylemmas 295
11.3 Themainresults 306
11.4 Comments 317
AppendixA Estimatesfortheheatkernel 319
AppendixB Heatpotentialestimates 328
AppendixC Nonlinearpotentialestimates 338
References 341
Index 348