This page intentionally left blank This page intentionally left blank PerturbationoftheBoundaryinBoundary-ValueProblems ofPartialDifferentialEquations LONDONMATHEMATICALSOCIETYLECTURENOTESERIES ManagingEditor:ProfessorN.J.Hitchin,MathematicalInstitute, UniversityofOxford,24-29StGiles,OxfordOX13LB,UnitedKingdom Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressatwww.cambridge.org 100 Stoppingtimetechniquesforanalystsandprobabilists, L.EGGHE 105 Alocalspectraltheoryforclosedoperators, I.ERDELYI&WANGSHENGWANG 107 CompactificationofSiegelmodulischemes, C.-L.CHAI 109 Diophantineanalysis, J.LOXTON&A.VANDERPOORTEN(eds) 113 Lecturesontheasymptotictheoryofideals, D.REES 116 Representationsofalgebras, P.J.WEBB(ed) 119 Triangulatedcategoriesintherepresentationtheoryoffinite-dimensionalalgebras, D.HAPPEL 121 ProceedingsofGroups-StAndrews1985, E.ROBERTSON&C.CAMPBELL(eds) 128 Descriptivesettheoryandthestructureofsetsofuniqueness, A.S.KECHRIS&A.LOUVEAU 130 Modeltheoryandmodules, M.PREST 131 Algebraic,extremal&metriccombinatorics, M.-M.DEZA,P.FRANKL&I.G.ROSENBERG(eds) 141 Surveysincombinatorics1989, J.SIEMONS(ed) 144 Introductiontouniformspaces, I.M.JAMES 146 Cohen-MacaulaymodulesoverCohen-Macaulayrings, Y.YOSHINO 148 Helicesandvectorbundles, A.N.RUDAKOVetal 149 Solitons,nonlinearevolutionequationsandinversescattering, M.ABLOWITZ&P.CLARKSON 150 Geometryoflow-dimensionalmanifolds1, S.DONALDSON&C.B.THOMAS(eds) 151 Geometryoflow-dimensionalmanifolds2, S.DONALDSON&C.B.THOMAS(eds) 152 Oligomorphicpermutationgroups, P.CAMERON 153 L-functionsandarithmetic, J.COATES&M.J.TAYLOR(eds) 155 Classificationtheoriesofpolarizedvarieties, TAKAOFUJITA 158 GeometryofBanachspaces, P.F.X.MU¨LLER&W.SCHACHERMAYER(eds) 159 GroupsStAndrews1989volume1, C.M.CAMPBELL&E.F.ROBERTSON(eds) 160 GroupsStAndrews1989volume2, C.M.CAMPBELL&E.F.ROBERTSON(eds) 161 Lecturesonblocktheory, BURKHARDKU¨LSHAMMER 163 Topicsinvarietiesofgrouprepresentations, S.M.VOVSI 164 Quasi-symmetricdesigns, M.S.SHRIKANDE&S.S.SANE 166 Surveysincombinatorics,1991, A.D.KEEDWELL(ed) 168 Representationsofalgebras, H.TACHIKAWA&S.BRENNER(eds) 169 Booleanfunctioncomplexity, M.S.PATERSON(ed) 170 ManifoldswithsingularitiesandtheAdams-Novikovspectralsequence, B.BOTVINNIK 171 Squares, A.R.RAJWADE 172 Algebraicvarieties, GEORGER.KEMPF 173 Discretegroupsandgeometry, W.J.HARVEY&C.MACLACHLAN(eds) 174 Lecturesonmechanics, J.E.MARSDEN 175 Adamsmemorialsymposiumonalgebraictopology1, N.RAY&G.WALKER(eds) 176 Adamsmemorialsymposiumonalgebraictopology2, N.RAY&G.WALKER(eds) 177 Applicationsofcategoriesincomputerscience, M.FOURMAN,P.JOHNSTONE&A.PITTS(eds) 178 LowerK-andL-theory, A.RANICKI 179 Complexprojectivegeometry, G.ELLINGSRUDetal 180 LecturesonergodictheoryandPesintheoryoncompactmanifolds, M.POLLICOTT 181 GeometricgrouptheoryI, G.A.NIBLO&M.A.ROLLER(eds) 182 GeometricgrouptheoryII, G.A.NIBLO&M.A.ROLLER(eds) 183 Shintanizetafunctions, A.YUKIE 184 Arithmeticalfunctions, W.SCHWARZ&J.SPILKER 185 Representationsofsolvablegroups, O.MANZ&T.R.WOLF 186 Complexity:knots,colouringsandcounting, D.J.A.WELSH 187 Surveysincombinatorics,1993, K.WALKER(ed) 188 Localanalysisfortheoddordertheorem, H.BENDER&G.GLAUBERMAN 189 Locallypresentableandaccessiblecategories, J.ADAMEK&J.ROSICKY 190 Polynomialinvariantsoffinitegroups, D.J.BENSON 191 Finitegeometryandcombinatorics, F.DECLERCKetal 192 Symplecticgeometry, D.SALAMON(ed) 194 Independentrandomvariablesandrearrangementinvariantspaces, M.BRAVERMAN 195 Arithmeticofblowupalgebras, WOLMERVASCONCELOS 196 Microlocalanalysisfordifferentialoperators, A.GRIGIS&J.SJO¨STRAND 197 Two-dimensionalhomotopyandcombinatorialgrouptheory, C.HOG-ANGELONIetal 198 Thealgebraiccharacterizationofgeometric4-manifolds, J.A.HILLMAN 199 InvariantpotentialtheoryintheunitballofCn, MANFREDSTOLL 200 TheGrothendiecktheoryofdessinsd’enfant, L.SCHNEPS(ed) 201 Singularities, JEAN-PAULBRASSELET(ed) 202 Thetechniqueofpseudodifferentialoperators, H.O.CORDES 203 HochschildcohomologyofvonNeumannalgebras, A.SINCLAIR&R.SMITH 204 Combinatorialandgeometricgrouptheory, A.J.DUNCAN,N.D.GILBERT&J.HOWIE(eds) 205 Ergodictheoryanditsconnectionswithharmonicanalysis, K.PETERSEN&I.SALAMA(eds) 207 GroupsofLietypeandtheirgeometries, W.M.KANTOR&L.DIMARTINO(eds) 208 Vectorbundlesinalgebraicgeometry, N.J.HITCHIN,P.NEWSTEAD&W.M.OXBURY(eds) 209 Arithmeticofdiagonalhypersurfacesoverfinitefields, F.Q.GOUVE´A&N.YUI 210 HilbertC∗-modules, E.C.LANCE 211 Groups93Galway/StAndrewsI, C.M.CAMPBELLetal(eds) 212 Groups93Galway/StAndrewsII, C.M.CAMPBELLetal(eds) 214 GeneralisedEuler-Jacobiinversionformulaandasymptoticsbeyondallorders, V.KOWALENKOetal 215 Numbertheory1992–93, S.DAVID(ed) 216 Stochasticpartialdifferentialequations, A.ETHERIDGE(ed) 217 Quadraticformswithapplicationstoalgebraicgeometryandtopology, A.PFISTER 218 Surveysincombinatorics,1995, PETERROWLINSON(ed) 220 Algebraicsettheory, A.JOYAL&I.MOERDIJK 221 Harmonicapproximation, S.J.GARDINER 222 Advancesinlinearlogic, J.-Y.GIRARD,Y.LAFONT&L.REGNIER(eds) 223 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems, KAZUAKITAIRA 224 Computability,enumerability,unsolvability, S.B.COOPER,T.A.SLAMAN&S.S.WAINER(eds) 225 Amathematicalintroductiontostringtheory, S.ALBEVERIO,etal 226 Novikovconjectures,indextheoremsandrigidityI, S.FERRY,A.RANICKI&J.ROSENBERG(eds) 227 Novikovconjectures,indextheoremsandrigidityII, S.FERRY,A.RANICKI&J.ROSENBERG(eds) 228 ErgodictheoryofZdactions, M.POLLICOTT&K.SCHMIDT(eds) 229 Ergodicityforinfinitedimensionalsystems, G.DAPRATO&J.ZABCZYK 230 Prolegomenatoamiddlebrowarithmeticofcurvesofgenus2, J.W.S.CASSELS&E.V.FLYNN 231 Semigrouptheoryanditsapplications, K.H.HOFMANN&M.W.MISLOVE(eds) 232 ThedescriptivesettheoryofPolishgroupactions, H.BECKER&A.S.KECHRIS 233 Finitefieldsandapplications, S.COHEN&H.NIEDERREITER(eds) 234 Introductiontosubfactors, V.JONES&V.S.SUNDER 235 Numbertheory1993–94, S.DAVID(ed) 236 TheJamesforest, H.FETTER&B.GAMBOADEBUEN 237 Sievemethods,exponentialsums,andtheirapplicationsinnumbertheory, G.R.H.GREAVESetal 238 Representationtheoryandalgebraicgeometry, A.MARTSINKOVSKY&G.TODOROV(eds) 240 Stablegroups, FRANKO.WAGNER 241 Surveysincombinatorics,1997, R.A.BAILEY(ed) 242 GeometricGaloisactionsI, L.SCHNEPS&P.LOCHAK(eds) 243 GeometricGaloisactionsII, L.SCHNEPS&P.LOCHAK(eds) 244 Modeltheoryofgroupsandautomorphismgroups, D.EVANS(ed) 245 Geometry,combinatorialdesignsandrelatedstructures, J.W.P.HIRSCHFELDetal 246 p-Automorphismsoffinitep-groups, E.I.KHUKHRO 247 Analyticnumbertheory, Y.MOTOHASHI(ed) 248 Tametopologyando-minimalstructures, LOUVANDENDRIES 249 Theatlasoffinitegroups:tenyearson, ROBERTCURTIS&ROBERTWILSON(eds) 250 Charactersandblocksoffinitegroups, G.NAVARRO 251 Gro¨bnerbasesandapplications, B.BUCHBERGER&F.WINKLER(eds) 252 Geometryandcohomologyingrouptheory, P.KROPHOLLER,G.NIBLO&R.STO¨HR(eds) 253 Theq-Schuralgebra, S.DONKIN 254 Galoisrepresentationsinarithmeticalgebraicgeometry, A.J.SCHOLL&R.L.TAYLOR(eds) 255 Symmetriesandintegrabilityofdifferenceequations, P.A.CLARKSON&F.W.NIJHOFF(eds) 256 AspectsofGaloistheory, HELMUTVO¨LKLEINetal 257 Anintroductiontononcommutativedifferentialgeometryanditsphysicalapplications2ed, J.MADORE 258 Setsandproofs, S.B.COOPER&J.TRUSS(eds) 259 Modelsandcomputability, S.B.COOPER&J.TRUSS(eds) 260 GroupsStAndrews1997inBath,I, C.M.CAMPBELLetal 261 GroupsStAndrews1997inBath,II, C.M.CAMPBELLetal 262 Analysisandlogic, C.W.HENSON,J.IOVINO,A.S.KECHRIS&E.ODELL 263 Singularitytheory, BILLBRUCE&DAVIDMOND(eds) 264 Newtrendsinalgebraicgeometry, K.HULEK,F.CATANESE,C.PETERS&M.REID(eds) 265 Ellipticcurvesincryptography, I.BLAKE,G.SEROUSSI&N.SMART 267 Surveysincombinatorics,1999, J.D.LAMB&D.A.PREECE(eds) 268 Spectralasymptoticsinthesemi-classicallimit, M.DIMASSI&J.SJO¨STRAND 269 Ergodictheoryandtopologicaldynamics, M.B.BEKKA&M.MAYER 270 AnalysisonLiegroups, N.T.VAROPOULOS&S.MUSTAPHA 271 Singularperturbationsofdifferentialoperators, S.ALBEVERIO&P.KURASOV 272 Charactertheoryfortheoddordertheorem, T.PETERFALVI 273 Spectraltheoryandgeometry, E.B.DAVIES&Y.SAFAROV(eds) 274 TheMandlebrotset,themeandvariations, TANLEI(ed) 275 Descriptivesettheoryanddynamicalsystems, M.FOREMANetal 276 Singularitiesofplanecurves, E.CASAS-ALVERO 277 Computationalandgeometricaspectsofmodernalgebra, M.D.ATKINSONetal 278 Globalattractorsinabstractparabolicproblems, J.W.CHOLEWA&T.DLOTKO 279 Topicsinsymbolicdynamicsandapplications, F.BLANCHARD,A.MAASS&A.NOGUEIRA(eds) 280 CharactersandautomorphismgroupsofcompactRiemannsurfaces, THOMASBREUER 281 Explicitbirationalgeometryof3-folds, ALESSIOCORTI&MILESREID(eds) 282 Auslander-Buchweitzapproximationsofequivariantmodules, M.HASHIMOTO 283 Nonlinearelasticity, Y.FU&R.W.OGDEN(eds) 284 Foundationsofcomputationalmathematics, R.DEVORE,A.ISERLES&E.SU¨LI(eds) 285 Rationalpointsoncurvesoverfinite,fields, H.NIEDERREITER&C.XING 286 Cliffordalgebrasandspinors2ed, P.LOUNESTO 287 TopicsonRiemannsurfacesandFuchsiangroups, E.BUJALANCEetal 288 Surveysincombinatorics,2001, J.HIRSCHFELD(ed) 289 AspectsofSobolev-typeinequalities, L.SALOFF-COSTE 290 QuantumgroupsandLietheory, A.PRESSLEY(ed) 291 Titsbuildingsandthemodeltheoryofgroups, K.TENT(ed) 292 Aquantumgroupsprimer, S.MAJID 293 SecondorderpartialdifferentialequationsinHilbertspaces, G.DAPRATO&J.ZABCZYK 294 Introductiontothetheoryofoperatorspaces, G.PISIER 295 GeometryandIntegrability, LIONELMASON&YAVUZNUTKU(eds.) 296 Lecturesoninvarianttheory, IGORDOLGACHEV 297 Thehomotopycategoryofsimplyconnected4-manifolds, H.-J.BAUES 299 KleinianGroupsandHyperbolic3-Manifolds Y.KOMORI,V.MARKOVIC,C.SERIES(eds.) 300 IntroductiontoMo¨biusDifferentialGeometry, UDOHERTRICH-JEROMIN 301 StableModulesandtheD(2)-Problem, F.E.A.JOHNSON 302 DiscreteandContinuousNonlinearSchro¨dingerSystems, M.J.ABLORWITZ,B.PRINARI,A.D.TRUBATCH 303 NumberTheoryandAlgebraicGeometry, MILESREIDALEXEISKOROBOGATOV(eds.) 304 GroupsStAndrews2001inOxfordVol.1, COLINCAMPBELL,EDMUNDROBERTSON,GEOFFSMITH(eds.) 305 GroupsStAndrews2001inOxfordVol.2, C.M.CAMPBELL,E.F.ROBERTSON,G.C.SMITH(eds.) 307 SurveysinCombinatorics2003, C.D.WENSLEY(ed.) 309 CoringsandComodules, TOMASZBRZEZINSKI&ROBERTWISBAURER 310 TopicsinDynamicsandErgodicTheory, SERGEYBEZUGLYI&SERGIYKOLYADA(eds.) 312 FoundationsofComputationalMathematics,Minneapolis2002, FELIPECUCKERetal(eds.) LondonMathematicalSocietyLectureNoteSeries.318 Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations DanHenry witheditorialassistancefrom JackHale and AntoˆnioLuizPereira    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press TheEdinburghBuilding,Cambridge,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridg e.org /9780521574914 © Cambridge University Press 2005 Thisbookisincopyright.Subjecttostatutoryexceptionandtotheprovisionof relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace withoutthewrittenpermissionofCambridgeUniversityPress. Firstpublishedinprintformat 2005 - ---- eBook (MyiLibrary) - --- eBook (MyiLibrary) - ---- paperback - --- paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof sforexternalorthird-partyinternetwebsitesreferredtointhisbook,anddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. Contents Introduction page1 1 GeometricalPreliminaries 3 1.1 Somenotation 3 2 DifferentialCalculusofBoundaryPerturbations 18 3 ExamplesUsingtheImplicitFunctionTheorem 27 3.1 TorsionalRigidity 27 3.2 SimpleEigenvaluesoftheDirichletProblemfortheLaplacian 32 3.3 Capacity 36 3.4 Green’sFunction 39 3.5 SimpleEigenvaluesofRobin’sProblem 40 3.6 SimpleEigenvalueofaGeneralDirichletProblem 44 4 BifurcationProblems 46 4.1 MultipleEigenvaluesoftheDirichletProblemfor theLaplacian 46 4.2 VariationofaTurningPoint 51 4.3 ABifurcationProblemwitTwo-DimensionalKernel 53 4.4 GenericSimplicityofEigenvaluesofaSelf-Adjoint2m-Order DirichletProblem 55 5 TheTransversalityTheorem 60 6 GenericPerturbationoftheBoundary 79 6.1 GenericSimplicityofEigenvaluesoftheDirichlet Problemfor(cid:1) 80 vii viii Contents 6.2 SimplicityofEigenvalueswithaReflection-symmetry Constraint 83 6.3 SimplicityofRealEigenvaluesofaGeneralSecond-Order DirichletProblem 85 6.4 GenericSimplicityofEigenvaluesfortheNeumann Problemfor(cid:1) 88 6.5 GenericSimplicityof(cid:1)u+ f(x,u,∇u)=0in(cid:2),u =0 on∂(cid:2) 89 6.6 GenericSimplicityofEigenvaluesofRobin’sProblem 94 6.7 GenericSimplicityofSolutionsof(cid:1)u+ f(u,x)=0in(cid:2), ∂u/∂N =g(k,u)on∂(cid:2) 100 6.8 GenericSimplicityofComplexEigenvaluesofaDirichlet Problem 106 7 BoundaryOperatorsforSecond-OrderElliptic Equations 114 7.1 WeaklySingularIntegralOperatorswithCr(α)Kernels 116 ∗ 7.2 IntegralEquationswithCr(α)Kernels 122 ∗ 7.3 FourierTransformandCompositionofWeakly SingularKernels 126 7.4 LimitsofSingularIntegrals 135 7.5 FundamentalSolutions 139 7.6 CalculationofsomeboundaryOperators 141 8 TheMethodofRapidly-OscillatingSolutions 152 8.1 Introduction 152 8.2 FormalAsymptoticSolutions 154 8.3 ExactSolutions 158 8.4 Example6.8Revisited:GenericSimplicityof ComplexEigenvalues 161 8.5 GenericSimplicityofSolutionsofaSystem 168 Appendix1 EigenvaluesoftheLaplacianinthePresence ofSymmetry 183 Appendix2 OnMicheletti’sMetricSpace 188 References 199 Index 203