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 LibraryofCongressCataloging-in-PublicationData ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress. BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2016WalterdeGruyterGmbH,Berlin/Boston Typesetting:PTP-Berlin,Protago-TEX-ProductionGmbH,Berlin Printingandbinding:CPIbooksGmbH,Leck ♾Printedonacid-freepaper PrintedinGermany www.degruyter.com 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