The Hodge-Laplacian: Boundary Value Problems on Riemannian Manifolds

528 Pages·2016·2.739 MB·English

by  Dorina Mitrea, Irina Mitrea, Marius Mitrea, Michael Taylor

DorinaMitrea,IrinaMitrea,MariusMitrea,andMichaelTaylor
TheHodge-Laplacian
De Gruyter Studies
in Mathematics
|
Editedby
CarstenCarstensen,Berlin,Germany
FarkasGavril,Berlin,Germany
NicolaFusco,Napoli,Italy
FritzGesztesy,Waco,Texas,USA
NielsJacob,Swansea,UnitedKingdom
ZenghuLi,Beijing,China
Karl-HermannNeeb,Erlangen,Germany
Volume 64
Dorina Mitrea, Irina Mitrea, Marius Mitrea,
and Michael Taylor
The Hodge-
Laplacian
|
Boundary Value Problems on Riemannian Manifolds
MathematicsSubjectClassification2010
31B10,31B25,31C12,35A01,35B20,35J08,35J25,35J55,35J57,35Q61,35R01,42B20,42B25,
42B37,45A05,45B05,45E05,45F15,45P05,47B38,47G10,49Q15,58A10,58A12,58A14,58A15,
58A30,58C35,58J05,58J32,78A30
Authors
DorinaMitrea IrinaMitrea
DepartmentofMathematics DepartmentofMathematics
UniversityofMissouriatColumbia TempleUniversity
Columbia,MO65211,USA Philadelphia,PA19122,USA
e-mail:[email protected] e-mail:[email protected]
MariusMitrea MichaelTaylor
DepartmentofMathematics DepartmentofMathematics
UniversityofMissouriatColumbia UniversityofNorthCarolina
Columbia,MO65211,USA ChapelHill,NC27599-3250,USA
e-mail:[email protected] e-mail:[email protected]
ISBN978-3-11-048266-9
e-ISBN(PDF)978-3-11-048438-0
e-ISBN(EPUB)978-3-11-048339-0
Set-ISBN978-3-11-048439-7
ISSN0179-0986
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Preface
ThismonographisdevotedtoanaturalclassofboundaryproblemsfortheHodge-
Laplacian,actingondifferentialforms.Thisclassincludestheabsoluteandrelative
boundaryconditions,usedintheHodge-stylerepresentationofabsoluteandrelative
cohomologyclassesoftheunderlyingdomainbyharmonicforms.
Continuingtheprogramin[86]aimedatunderstandingthesolvabilityproperties
ofsuchboundaryproblemsunderminimalgeometricandanalyticregularityassump-
tions,herewepushfurthertheanalysisdevelopedin[50]ofalayerpotentialattack
onellipticboundaryproblemsonaclassofdomainsintroducedbySemmes[112]and
KenigandToro[66],whichwecallregularSemmes-Kenig-Toro(SKT)domains. We
initiatethestudyofboundaryvalueproblemsfordifferentialformsinthisclassofdo-
mains.Inadditiontotheabsoluteandrelativeboundaryconditionsmentionedearlier,
wealsotreattheHodge-LaplacianequippedwithclassicalDirichlet,Neumann,Trans-
mission,Poincaré,andRobinboundaryconditionsinregularSKTdomains,withdata
inLpspaces,forarbitraryp∈(1,∞).
Inabroadperspective,ourresultsmayberegardedasanaturalcompletion,of
anoptimalnature,oftheworkinitiatedbyE.Fabes,M.Jodeit,andN.Rivièrein[32],
whosescopeisextendedherethroughtheconsiderationofdifferentialformsinplace
ofscalarfunctions,the(variable-coefficient)Hodge-Laplacianinlieuofthe(constant
coefficient)Laplaceoperator,andregularSKT subdomainsofRiemannianmanifolds,
witharbitrarytopology,replacingC1domainswithconnectedcompactboundariesin
theflatEuclideansetting.
Instarkcontrasttothescalarcasefrom[32],thestructuralrichnessofthehigher
degreecaseconsideredhereallowsforamuchlargervarietyofnaturalboundaryvalue
problemsfortheHodge-Laplacian,whichweformulateandstudysystematicallyvia
potentialtheoreticmethods.
DorinaMitrea,Columbia,MO,USA
IrinaMitrea,Philadelphia,PA,USA
MariusMitrea,Columbia,MO,USA
MichaelTaylor,ChapelHill,NC,USA

Contents
Preface|v
1 IntroductionandStatementofMainResults|1
1.1 FirstMainResult:AbsoluteandRelativeBoundaryConditions|3
1.2 OtherProblemsInvolvingTangentialandNormalComponentsof
HarmonicForms|11
1.3 BoundaryValueProblemsforHodge-DiracOperators|21
1.4 Dirichlet,Neumann,Transmission,Poincaré,andRobin-TypeBoundary
Problems|24
1.5 StructureoftheMonograph|43
2 GeometricConceptsandTools|49
2.1 DifferentialGeometricPreliminaries|49
2.2 ElementsofGeometricMeasureTheory|67
2.3 SharpIntegrationbyPartsFormulasforDifferentialFormsinAhlfors
RegularDomains|91
2.4 TangentialandNormalDifferentialFormsonAhlforsRegularSets|96
3 HarmonicLayerPotentialsAssociatedwiththeHodge-deRhamFormalism
onURDomains|109
3.1 AFundamentalSolutionfortheHodge-Laplacian|109
3.2 LayerPotentialsfortheHodge-Laplacian
intheHodge-deRhamFormalism|117
3.3 FredholmTheoryforLayerPotentials
intheHodge-deRhamFormalism|128
4 HarmonicLayerPotentialsAssociatedwiththeLevi-CivitaConnection
onURDomains|139
4.1 TheDefinitionandMappingPropertiesoftheDoubleLayer|140
4.2 TheDoubleLayeronURSubdomainsofSmoothManifolds|169
4.3 CompactnessoftheDoubleLayeronRegularSKTDomains|173
5 DirichletandNeumannBoundaryValueProblemsfortheHodge-Laplacian
onRegularSKTDomains|185
5.1 FunctionalAnalyticPropertiesforHarmonicLayerPotentials
inURDomains|186
5.2 InvertibilityResultsforLayerPotentialsAssociated
withtheLevi-CivitaConnection|196
5.3 SolvingtheDirichlet,Neumann,Transmission,Poincaré,andRobin
BoundaryValueProblems|204
viii | Contents
6 FatouTheoremsandIntegralRepresentationsfortheHodge-Laplacian
onRegularSKTDomains|231
6.1 ConvergenceofFamiliesofSingularIntegralOperators|231
6.2 AFatouTheoremfortheHodge-Laplacian
inRegularSKTDomains|250
6.3 SpacesofHarmonicFieldsandGreenTypeFormulas|261
7 SolvabilityofBoundaryProblemsfortheHodge-Laplacian
intheHodge-deRhamFormalism|275
7.1 PreparatoryResults|275
7.2 SolvabilityResults|288
8 AdditionalResultsandApplications|315
8.1 deRhamCohomologyonRegularSKTSurfaces|315
8.2 Maxwell’sEquationsinRegularSKTDomains|336
8.3 Dirichlet-to-NeumannOperatorsfortheHodge-Laplacian
inRegularSKTDomains|339
8.4 FatouTypeResultswithAdditionalConstraints
orRegularityConditions|347
8.5 WeakTangentialandNormalTracesinRegularSKTDomains
withFriedrichsProperty|352
8.6 TheHodge-PoissonKernelandtheHodge-HarmonicMeasure|367
9 FurtherToolsfromDifferentialGeometry,HarmonicAnalysis,
GeometricMeasureTheory,FunctionalAnalysis,
PartialDifferentialEquations,andCliffordAnalysis|371
9.1 ConnectionsandCovariantDerivativesonVectorBundles|371
9.2 TheExtensionoftheLevi-CivitaConnectiontoDifferentialForms|381
9.3 TheBochner-LaplacianandWeintzenböck’sFormula|386
9.4 SobolevSpacesonBoundariesofAhlforsRegularDomains:
TheEuclideanSetting|393
9.5 SobolevSpacesonBoundariesofAhlforsRegularDomains:
TheManifoldSetting|408
9.6 IntegratingbyPartsontheBoundaries
ofAhlforsRegularDomains|417
9.7 AGlobalSobolevRegularityResult|444
9.8 ThePVHarmonicDoubleLayeronaURDomain|446
9.9 Calderón-ZygmundTheoryonURDomainsonManifolds|451
9.10 TheFredholmnessandInvertibilityofElliptic
DifferentialOperators|474
9.11 CompactandClose-to-CompactSingularIntegralOperators|482
9.12 ASharpDivergenceTheorem|490
Contents | ix
9.13 CliffordAnalysisRudiments|493
9.14 SpectralTheoryforUnboundedLinearOperators
SubjecttoCancellations|496
Bibliography|501
Index|507