On a model boundary value problem for Laplacian wi

arXiv:math-ph/0208009v322Mar2003OnamodelboundaryvalueproblemforLaplacianwithfrequentlyalternatingtypeofboundaryconditionDenisI.BorisovTheBashkirStatePedagogicalUniversity,OctoberRev.St,3a,450000,Ufa,Russiatel.7-3472-317827,fax[phone]E-mail:[e-mail],[e-mail]-dimensionalsingularperturbedeigenvalueproblemforLaplacianwithfrequentlyalternatingtypeofboundaryconditionisconsidered.Completetwo-parametricalasymptoticsfortheeigenelementsareconstructed.IntroductionEllipticboundaryvalueproblemswithfrequentlyalternatingtypeofbound-aryconditionaremathematicalmodelsusedinvariousapplications.Webrieflydescribetheformulationoftheseproblems.Inagivenboundeddomainwithasmoothorapiecewisesmoothboundaryanellipticequationisconsidered.Ontheboundaryoneselectsasubsetdependingonasmallparameterandconsistingofalargenumberofdisjointparts.Themeasureofeachparttendstozeroasthesmallparametertendstozero,whilethenumberofthesepartsincreasesinfinitely.OnthesubsetdescribedtheDirichletboundaryconditionisimposed,whereastheNeumannboundaryconditionisimposedontherestpartoftheboundary.Thereisanumberofpapersdevotedtoaveragingofsuchproblems(see,forinstance,[1]–[4]).Themainobjectiveoftheseworkswastodescribelimiting(homogenized)problems.Thecaseofperiodicalternatingofboundaryconditionswasinvestigatedin[2],[3],whilethenonperiodiconewastreatedin[1],[4].Themainresultoftheseworkscanbeformulatedasfollows.Theformoflimitingproblem(namely,typeofboundarycondition)dependsoftherelationbetweenmeasuresofpartsoftheboundarywithdifferenttypesofboundaryconditions.Furtherstudyingoftheboundaryvalueproblemswithfrequentlyalternatingboundaryconditionswascarriedoutintwodirections.Firstdirectionconsistsintheestimatesfordegreeofconvergenceunderminimalnumberofrestrictionstothestructureofalternatingofboundaryconditions([2],[4]–[6]).Anotherdirectioninstudyingoftheseproblemsisaconstructingtheasymptoticsexpansionsofsolutions.Presentpaperdevelopsexactlythisdirection.Inthispaperwestudyatwo-dimensionalsingularperturbedeigenvalueprob-lemforLaplaceoperatorinaunitcircleDwithcenterattheorigin.OntheboundaryofthecircleDweselectaperiodicsubsetγεconsistingofNdisjointarcs,lengthofeacharcequals2εη,whereN≫1isanintegernumber,ε=2N−1,η=η(ε),0ηπ/2.Eachofthesearcscanbeobtainedfromanneighbouringonebyrotationabouttheoriginthroughtheangleεπ(cf.figure).OnγεweimposetheDirichletboundaryconditionandtheNeumannboundaryconditionisconsid-eredontherestpartoftheboundary.From[1],[2]itfollowsthatthemainroleindeterminationoflimitingproblembelongstothelimitlimε→0(εlnη(ε))−1=−A.IfA≥0,thenthelimitingproblemiseithertheRobinproblem(A0)ortheNeu-mannproblem(A=0).Theassumptionlimε→0(εlnη(ε))−1=−Adoesnotdefinethefunctionη(ε)uniquely;clear,itisequivalenttotheequalityη(ε)=exp−1ε(A+μ),whereμ=μ(ε)isanarbitraryfunctiontendingtozeroasε→0,andalso,A+μ0forε0.Thus,theproblemstudiedcontainsactuallytwoparameters,εandμ.Inpaper[7]completepower(onε)asymptoticsfortheeigenelementsoftheperturbedproblemwereconstructedinthecaseoftheNeumannlimitingproblem(A=0)underanadditionalassumptionμ(ε)=A0ε,A0=const0.InthispaperwestudythecaseoflimitingNeumannorRobinproblem(A≥0)withoutanyadditionalassumptionsforη(ε).OnthebasisofthemethodFigure.ofmatchedasymptoticsexpansions[8],themethodofcompositeexpansions[9]andthemultiscaledmethod[10]weobtaincompletetwo-parametrical(onεandμ)asymptoticsfortheeigenelementsoftheperturbedproblem.Employingtheasymptoticsexpansionsfortheeigenvalues,weprovethattheperturbedproblemhasonlysimpleanddoubleeigenvalues,andweshowcriteriondistinguishingthesecases.1.TheproblemandmainresultsLetx=(x1,x2)betheCartesiancoordinates,(r,θ)betheassociatedpolarcoordinates,Γε=∂D\γε.WithoutlossofgeneralitywemayassumethatthesetγεissymmetricwithrespecttotheaxisOx1.Westudysingularperturbedeigenvalueproblem−Δψε=λεψε,x∈D,(1.1)ψε=0,x∈γε,∂ψε∂r=0,x∈Γε.(1.2)¿From[1],[2]itfollowsthatinthecaseA≥0theeigenelementsoftheperturbedproblemconvergetotheeigenelementsofthefollowinglimitingproblem−Δψ0=λ0ψ0,x∈D,∂∂r+Aψ0=0,x∈∂D.(1.3)TheeigenfunctionsconvergestronglyinL2(D)andweaklyinH1(D).Totalmul-tiplicityoftheperturbedeigenvaluesconvergingtoap-multiplyeigenvalueequalsp.Itiswellknownfactthattheeigenvaluesoftheproblem(1.3)coincidewiththerootsoftheequationpλ0J′npλ0+AJnpλ0=0,(1.4)whereJnareBesselfunctionsofintegerordern≥0,andassociatedeigen-functionsaredefinedbytheequalitiesψ0=J0√λ0r
(forn=0)andψ±0=Jn√λ0r
φ±(nθ)(forn0),φ+=cos,φ−=sin.Remark1.1.Itshouldbestressedthattheproblem(1.3)canhaveeigenvaluesofvariousmultiplicity,includingmultiplicitymorethantwo.ThissituationtakesplacebecauseforsomevaluesofAthereexistsλ0beingrootofequation(1.4)fordifferentnsimultaneously.TheproofofexistenceofsuchAisgiveninAppendix.Thispaperisdevotedtotheproofofthefollowingstatement.Theorem1.1.Letλ0bearootoftheequation(1.4)forn≥0.Thenthereexistsaneigenvalueλεoftheperturbedproblemconvergingtoλ0andsatisfyingasymptoticsλε=Λ0(μ)+M−1Xi=3εiΛi(μ)+O(εM(A+μ)),(1.5)foranyM≥3,whereΛ0(μ)istherootoftheequationpΛ0J′npΛ0+(A+μ)JnpΛ0=0,Λ0(0)=λ0,(1.6)Λ3(μ)=−ζ(3)4(A+μ)2(Λ0(μ