Asymptotic behaviour of solutions of fourth order

Rend.Istit.Mat.Univ.TriesteVol.XXIX,19{38(1997)AsymptoticbehaviourofsolutionsoffourthorderDirichletproblemsPaoloDall’Aglio()Summary.-Theasymptoticbehaviourofsolutionstofourthor-derDirichletellipticproblems,onvaryingdomains,isstudiedthroughthedecompositionintoasystemofsecondorderones,whichleadstorelaxedformulationswiththeintroductionofmea-sureterms.Thisallowstosolveashapeoptimizationproblemforasimplysupportedthinplate.1.IntroductionInthispaperwestudytheasymptoticbehaviourofsolutionsoffourthorderellipticproblemsonvaryingdomains.Thishasbeenwidelystudiedinthepast,inthecaseofsecondorderellipticoperators(seeforinstance[6],[9],[7]).Wewillusesuchresultsdecomposingfourthorderdierentialequationsintoasystemofsecondorderones.GivenaboundedopensetUinIRn,n2andafunctionf2H-1(U),thefourthorderequation8:2u=finH-1(U)u2H10(U)u2H10(U)(1.1)islinkedtothemodelfortheverticaldisplacementuofanthinplate,occupyingaregionU,simplysupportedon@U,subjectedtoaloadf.Simplysupportedmeansthattheboundaryisxed,butthatthe()Author’saddress:S.I.S.S.A.,viaBeirut2/4,34013Trieste(Italy).20P.DALL’[e-mail]platetheorywereferto[15],[11],[4],[14],[5].Inparticularwewanttostudytheasymptoticbehaviourofsolu-tionswhenthedomainvaries.Tothisaimwewillshow(Proposition3.1)thatproblem(1.1)isequivalenttothesystemofsecondorderequations8:u=vinH-1(U)u2H10(U)v=finH-1(U)v2H10(U):(1.2)Thisproblemcanbehandledwiththetheoremsvalidinthesecondordercase(Theorems2.1and2.2),anditwillbeprovedthatifunarethesolutionsofproblemslike(1.2)onasequenceofsubdomainsUnofagivenboundeddomain,thenasubsequenceofunconvergesweaklyinH10()toafunctionusolving8:u+u=vu2H10()\L2()v+v=fv2H10()\L2();(1.3)whereisameasure.Thestudywillbecarriedonforgeneralfourthorderellipticop-eratorswithconstantcoecientsandnolowerorderterms,thatcanbesplitintotwosecondorderones.Amotivationforthestudyoftheasymptoticbehaviourofso-lutionsofDirichletproblemsinvaryingdomainswithoutgeometricassumptionsonthedomainsUnaretheso-calledshapeoptimizationproblems:givenafunctionj:IR!IRweconsiderthefollowingproblemminU2U()Zj(x;uU(x))dx;(1.4)whereU()isthefamilyofallopensubsetofanduUisthesolutionoftheproblemoftype(1.1)inthesetU.InSection6itwillbeshownthat,ingeneral,problem(1.4)doesnothaveasolution.HencearelaxedoptimizationproblemASYMPTOTICBEHAVIOUROFSOLUTIONSetc.21willbeintroduced,wherethesetoverwhichweminimizeisthesetoffunctionsu,whereisameasureandusolvestherelaxedproblem(1.3).ThissetistheclosureoffuU:U2U()ginH10()-weak.Thisproblemwillalwayshavesolutionanditsminimumwillcoincidewiththeinmumoftheintegralin(1.4).Thisshapeoptimizationproblem,inthesecondordercase,wasstudiedin[1],[2],[3].2.NotationsandpreliminaryresultsGivenanopensubsetUofIRn,H10(U)istheusualSobolevSpace,H-1(U)itsdual,andh;ithedualitypairing.IfUandu2H10(U),thenthefunction~u:=(uinU0innU;isinH10().Fromnowonwewillalwaysdenote,withthesamesymbolu,afunctionanditsextension~u.InthispaperwewilldealwithellipticoperatorsL:D0()!D0()oftheformLu=Xjj=mc@u;whereandaremultiindexes,andmistheorderoftheoperatorthatwillbe2or4.Inanycasetheywillbewithoutlowerordertermsandwithconstantcoecients.TheoperatorswillbeassumedtobeellipticinthesensethatXjj=mcjjm;82IRn;whereisarealpositiveconstant.This,inourcase,isthesameasP()6=082IRnnf0g;(2.1)wherePisthepolynomialXjj=mcassociatedtotheoperatorL.22P.DALL’AGLIOInthiswork,dierentialproblemsarealwaysmeanttobesolvedintheusualweaksense.Thismeans,forinstance,that,foru2H10(U)theexpressionu=finH-1(U)isanequalityoflinearfunctionalshu;vi=XiZU@iu@ivdx=hf;vi;foranyv2H10(U).Assaidabove,thelimitofasequenceofsolutionsofDirichletproblemsisnot,ingeneral,thesolutionofaproblemofthesamekind,butisthesolutionofaproblemwhereameasuretermappears.Todealwiththeseproblemsweneedtorecallsomenotions.ForthenotionofcapacityofasetE,whichwewillindicatebycap(E),werefertotextbooksas[12]or[13].Weshallalwaysidentifyafunctionu2H10()withitsquasi-continuousrepresenta-tive.Now,letM0()bethesetofBorelmeasureswhicharezeroonthesetsofzerocapacity.Foronesuchmeasure2M0(),L2(U)willbethespaceoffunctionssuchthatZUjuj2d+1:GivenasecondorderoperatorA=Xijaij@i@j,withconstantcoe-cients,forafunctionu2H10(U)\L2(U)tosolvetheequationAu+u=f;willmeanthatnXi=1ZUaij@ju@ivdx+ZUuvd=ZUfvdx;(2.2)foralltestfunctionsinv2H10(U)\L2(U).ASYMPTOTICBEHAVIOUROFSOLUTIONSetc.23ItcanbeeasilyprovedthatthespaceH10(U)\L2(U)isaHilbertspacewheneverisinM0(),andhence,byLax-MilgramLemma,wehaveexistenceanduniquenessofsolutionsforaproblemoftheform(Au+u=fu2H10(U)\L2(U);foranylinearellipticsecondorderoperatorA.Thedecompositionofafourthorderprobleminasystemoftwosecondorderequationsallowsustostudytheasymptoticbehaviourapplyingwellknowntheoremsforthesecondordercaseseparatelytoeachequation.Thefollowingresults,thatcanbefound,forinstancein[9]and[7],arethekeypointsofthetheory.Theorem2.1.LetAbeasecondorderellipticoperator,asde-scribedabove.ForeverysequencefnginM0()thereexistsasub-sequencenksuch