On the Boundary Integral Equation Method for a Mix

OntheBoundaryIntegralEquationMethodforaMixedBoundaryValueProblemoftheBiharmonicEquationFioralbaCakonia,1,GeorgeC.Hsiaoa,2andWolfgangL.Wendlandb,2aDepartmentofMathematicalSciences,UniversityofDelaware,Newark,Delaware19716,U.S.A.E-mail:{cakoni,hsiao}@math.udel.edubInstitutf¨urAngewandteAnalysisundNumerischeSimulation,Universit¨atStutgart,Pfaffenwaldring57,70569Stuttgart,GermanyE-mail:[e-mail]ehrontheoccasionofhis65thbirthdayAbstractThispaperisconcernedwithweaksolutionofamixedboundaryvalueproblemforthebiharmonicequationintheplane.UsingGreen’sformula,theproblemisconvertedintoasystemofFredholmintegralequationsfortheunknowndataondifferentpartoftheboundary.ExistenceanduniquenessofthesolutionsofthesystemofboundaryintegralequationsareestablishedinappropriateSobolevspaces.Keywords:Biharmonicequation,mixedboundaryvalueproblems,boundaryintegralequations,variationalformulations.1TheresearchofthisauthorwassupportedinpartbytheAirForceOfficeofScientificResearchundergrantF49620-02-1-0353.2TheresearchoftheseauthorsweresupportedinpartbytheGermanResearchFoundationDFGundertheGrantSFB404MultifieldProblemsinContinuumMe-chanicsPreprintsubmittedtoElsevierScience17February20051IntroductionInthepaper[14],amixedboundaryvalueproblemforthetwo-dimensionalLaplaceequationisconsidered.UsingGreen’sformulatheproblemiscon-vertedintoasystemofFredholmintegralequationsforthemissingpartofCauchydataondifferentpartoftheboundary.Oneoftheseboundaryintegralequationshasprincipalpartofthesecondkind,whereastheotheroneisofthefirstkind.However,thecrucialpointoftheapproachthereisthatthederivedsystemofintegralequationscanbeinterpretedasastronglyellipticsystemofpseudodifferentialequations.HenceitcanbesolvedconstructivelybyGalerkin’smethod.Thepurposeofthepresentpaperistoseethefeasibilityofextendingtheapproachin[14]fortheLaplaceequationtothebiharmonicequation.Clearlyforthelatter,itismuchmoreinvolved,becauseofthedoubleofCauchydata;thereare16boundaryoperatorsneededtobeconsidered.However,aswillbeseen,therecentsystematiccharacterizationoftheCalder`onprojectorin[8]hassimplifiedtheapproachinthesamemannerasinthecasefortheLaplaceequation.Thepaperisorganizedasfollows:InSection1,weformulatethemixedbound-aryproblemandpresentsomepreliminaryresultsfortheweaksolutionsoftheboundaryvalueproblemsforthebiharmonicequation.Section2containsthecorematerialsforthefourbasicboundaryintegralequations.Theorems3.3,3.4and3.5inSection3arethemainresultsconcerningexistenceanduniquenessofthesystemofboundaryintegralequationsinSection2.Finallyinthelastsection,Section4,weconcludethepaperbyabriefdiscussionontheregularityresultsofthesolutionsofboundaryintegralequations,andthesecanbeservedasthemathematicalfoundationsfortheaugmentedGalerkinmethodinthesamemannerasinthecaseofLaplaceequation(see[14]).2FormulationoftheProblemLetΩ⊂IR2beaboundedsimplyconnectedregionwithC1,1-boundaryΓ.WeassumethattheboundaryΓhasadissectionΓ=ΓD∪Γc∪ΓN,whereΓDandΓNaredisjoint,relativelyopensubsetofΓ,havingΠastheircommonboundaryinΓ.Wedenotebyn=(n1,n2)theunitoutwardnormalvectortoΓ.NowletathinplateinelastostaticequilibriumoccupytheregionΩ.WeassumethatthepartΓDoftheboundaryisclampedwhilethepartΓNisfree.Ifwedenotebyutheequilibriumstateoftheplateweobtainthefollowingmixed2boundaryvalueproblemsforthebiharmonicequationΔ2u=0inΩ(1)u=fand∂u∂n=gonΓD(2)Mu=pandNu=qonΓN,(3)wheretheboundaryoperatorsM|ΓDandN|ΓNaretherestrictiontoΓDandΓNofthefollowingboundarydifferentialoperatorsMu=νΔu+(1−ν)M0u(4)andNu=−∂∂nΔu−(1−ν)∂∂sN0u,(5)respectively.Hereνisarealconstant,thePoissonratioandinapplication(especiallyinthetheoryofelasticity)wehave0≤ν1.Thenormalandtangentialderivativesaregivenby∂∂n=n1∂∂x1+n2∂∂x2and∂∂s=−n1∂∂x2+n2∂∂x1,whiletheboundaryoperatorsM0uandN0uaredefinedbyM0u:=∂2u∂x21n21+2∂2u∂x1∂x2n1n2+∂2u∂x22n22andN0u:=−n∂2u∂x21−∂2u∂x22!n1n2−∂2u∂x1∂x2n21−n22o.Physically,MuisthebendingmomentandNuisthetransverseforceconsist-ingoftheshearforceandtwistingmoment[2].Themixedconditions(2)and(3)maybeinterpretedthattheplateisclampedonΓDandhasafreeedgeonΓN.Weareinterestedintheweaksolutionsofthemixedboundaryvalueproblem,(1),(2),and(3).Oursolutionspaceforthebiharmonicequation(1)isthestandardSobolevspaceH2(Ω)ofdistributionsthataresquareintegrableandhavesquarein-tegrablederivativesuptothesecondorder.WefirstobservethatsincetheboundaryΓisC1,1,thetracespacesH32(Γ)andH12(Γ)arewelldefined[?]andmoreoverforu∈H2(Ω)wehavethatu|Γ∈H32(Γ)and∂u∂nΓ∈H12(Γ).Todiscusstheboundaryvalueproblemfor(1),itisbesttobeginwiththe3Greenformulafor(1)inΩ.Byusingintegrationbypartsformulas,onecanshowthatthefollowingGreenformulaholdsZΩ(Δ2u)vdx=a(u,v)−ZΓ(Mu)∂v∂n+(Nu)v#ds,(6)forsmoothfunctions,wherethebilinearforma(u,v)isdefinedbyaΩ(u,v):=ZΩνΔuΔvdx+ZΩ(1−ν)∂2u∂x21∂2v∂x21+2∂2u∂x1∂x2∂2v∂x1∂x2+∂2u∂x22∂2v∂x22!dx.(7)Wenotethatthebilinearformin(7)iswelldefinedforfunctionsinH2(Ω).Nowletu∈H2(Ω,Δ2)whereH2(Ω,Δ2):={u∈H2(Ω):Δ2u∈fH−2(Ω)}withfH−2(Ω)denotingthedualspaceofH2(Ω)andchoosev∈H2(Ω).ThentheaboveGreenformulaholdsandbyadualityargumentoneshowsthatMu∈H−12(Γ)andNu∈H−32(Γ)arewelld