; TeX output 1999.01.01:2053N~MkN cmbx12kNenadAntonic&NevenBalenovic'W{Nff cmbx12{OptimalffdesignforplatesandrelaxationF(C̫]kAbstract W<CK`y cmr10CTheoptimaldesignproblemforaplateofvqariablethicknessassumingtheKirchho (moGdeltforpurebendingofsymmetricplatesisstudied.&Itiswellknownthatthisproblem(hasWnosolution.TherelaxationproGcedureisthusnecessary*,0WintroGducingcomposite(materials.qThisUUnaturallyleadstohomogenisation.<F*ollowingltheideasofTartar,anewproGcedureisproposed,givingtheknownresult(of 3 cmmi10THS-con!vergence,fYeoungmeasures|Departmen!tfofMathematics 33Univ!ersityfofZagrebBijeni"DckdDafcesta30Zagreb,fCroatia \T `[SnU[S S&^Mz1 \|S([xS)[SnU[S a>S1=ڟ㦉p ł T 6[SU[SS(ae [xS)uV AT:aSIndeed,ffor^M U2`Mz2S(T `;1 S; ),fthebilinearformh(5),TaS(Tu;1vdS) :=VZ  ^MUrrTunUrrTv7d[xoSisO2ellipticandbMoundedonH2S0S( )&T;fMz1n:j1222UTM1nS1211TM1nS1122㦉p 8? TM:u1nS1111L-:1R( );㍍eUԅz6蘹IT*GMz11:j1222UTM11S1211TM11S1122㦉p 8? TM:u11S1111>T;fMz1n:j2212UTM1nS2211TM1nS1112㦉p 8? TM:u1nS1111L-:1R( );㍍eUԅz6蘹IT*GMz11:j2212UTM11S2211TM11S1112㦉p 8? TM:u11S1111>T;fMz1n:j2222UTM1nS2211TM1nS1122㦉p 8? TM:u1nS1111L-:1R( );㍍eUԅz6蘹IT*GMz11:j2222UTM11S2211TM11S1122㦉p 8? TM:u11S1111>T:ZIf@TuznPZ,$foreac!hTn^U2[N|U[f1gZ,is@thesolutionofproblem(1,4)correspMondingto^MnPZ,thenTuznHbBH-:2N0*( )Ud !6#wK*(`T*8&RWee 33sa!y$thatasequenceoftensorfunctions(^MnPS)in`Mz2S(T `;1 S; )$THS-convergesto^M1 U2`Mz2S(T `;1 S; )iffforan!yTf8cU2 SH2 \|S( )*thesequenceofsolutions(TuznPS)oftheproblems#ύ(7)fOwV8 獑Ow<ύOw: [SFindfTuznU2 SH2S0S( )$Gsuc!hthatgx(U81Tvo:U2 SHz2:j0S( ) m)᠐VZ ,^MznPUrrTuzn2N0*( )>T:SAstheformTaSisbMoundedandcoerciv!e,itfollowsthattheopMeratorTASisboundedandin!vertible, 33andthatitsin!verseTA1Siscon!tinuousasw!ell.EThusweareabletoapplythestandardresultsvdDalidfor0gtheTGS-con!vergence0gofopMerators[;jZK!ON],whichgiveusappropriatecompactness(moreover,forsymmetrictensors^Mn nSw!ehavethecompactnesswiththesameconstants).Ofcourse,^ itremainstobMepro!venthattheoperatorbeingtheTGS-limitoftheoperatorsoftheform(8),isofthefsameformitself.r[LemmaJ3.KZLetTVsZbMereal,re exiv!eandseparableBanachspace.IftheopMeratorTA>U2LS(TVnS;1TV0=(S)ZisݟcoMerciv!e(i.e.S(U91T q>fS0)(U8TuU2TVnS)UhTAu;uUia>T `UkTuUkEBJ2HBJVZ),mthenݟtheequationTAuS=Tf [ZhastheuniquesolutionfTu U2TVUZforan!yTf8cU2TVn0=(Z,andtheinequalit!y5퍍BUkTuUk, (U81TuU2SHz2:j0S( ) m) 5UhTAznTu;1uUia>T `UkTuUkBJ2N0*( ),T:@SOurgoalistopro!vethecompactnessoftheset`Mz2S(T `;1 S; );#asthe rststepw!eachieve,IbMesidesthe+pTGS-con!vergenceoftheopMerators,Dtheabstractconvergenceofthesequence(^MnPUrrTuznS)asw!ell.ThatflimitwillbMeiden!ti edinthenextstep.[Lemma7'4.KZTheregisasubsequenceS(^Mni? ;cmmi6k ?jS)ZoftheabMo!vegsequence,andtheoperatorsTAz1aU2LS(H2S0S( )c:;1H2 \|S( )%)YZandfTR OU2 LS(H2 \|S( )#;1L2S( ;Sym)7)uZsuc!hthat,HTAzni?krG5 ?jUR!)ETAz1Z(i.e.fTA1Ani?kg#U8T* A1A1Zw!eaklyinthesenseofopMerators),andthatforarbitraryTf8cU2SH2 \|S( )*Zw!ehave٪^Mzni?k ?jUrrTuzni?k L-:2*( ;SymJ);㍍ JU&;!]P(e.zT*2N0*( )H;L2*( ;SymJ)()(a6 T n:ƖZThenf^M1 U2 SL1 S( ;1ULS(SymT;SSym))jZandw!ehaveUj^Mz1 S([xS)Uj:uL(SymJ;Sym)2a6 T %Sss"[xU2S 1T: ɦLH!kq^k3.E ectiveprop`ertiesofalayeredplateSThe|goalofthissectionistopro!ve|Lemma1,whic!henablesustogeneralisetheresultofMuM~8nozandP!edregal[MP]}toamoregeneralsetting.QInfact,VLemma1isasimpleconsequenceoffthefollo!wing[Theorem3.KZLetS U[R2ZbMeanopenandboundedset,2andS(^MnPS)ZasequenceoftensorvdDaluedfunctions!]Ti S=TjS=133 TDzMn:jij T;-SotherwisepcxT:#SCom!biningfthiswith(9)weobtainYab[OznS= ^KznP[Gzn{T;YaSwhere^Kn 5S:= (^MnPS).ɷReasonablybriefcalculationsho!wsthatthecompMonentsoftensor^Kn TSare(uptoaconstan!t)exactlythetermsonthelefthandsidein(6).QOntheotherhand,since^MnSbMelongsStothespace`Mz2S(T `;1 S; ),w!eSconcludethatT a6 TM1nS1111S([xS)a6T 9S(ae [xU2S ),whic!hSclearlyimplies1}thatthesequence(^KnPS)isbMoundedinL1 S( ;1ULS(Sym;Sym)@b)g=b,Handhencehasaclusterpoin!tinfw!eak{UStopMologye.Usingthefactthat([GnPS)doMesnotoscillateinTxz1S,sand(^KnS)depMendonTxz1Sonlye,saftertakingafsubsequence,w!eobtainYa_z1 S= ^Kz1 [Gz1 9T:YaSByfusingthisequalit!yitiseasytoreado that^K1 S:= (^M1 S),orinotherwordsYal (^Mzni?k ?jS) ,L-:1R( )$;㍍ U l$T*2}S (^Mz1 S)1T;Swhic!hEisexactlywhatwewantedtoprove.Ofcourse,Y-eachclusterpMointofthesequence^Kn/Smustsatisfy#theabMo!ve#convergenceresult,Rthusitholdsfortheentiresequence( (^MnPS)),Rratherthanforfasubsequence.ThisfurnishestheproMofofZonlyifSpartofthetheorem. ꘍AsffortheZifSpart,w!eareinfactabletoproveaslightlystrongerresultYa[Theorem4.KZWithnotationasintheproMofofTheorem3,SassumethatthesequenceS(TuznPS)Zcon-v!ergesxweaklyinSH2S( )%f*ZtoTuz1 Z,TM1nS1111S([xS)a6T fS(ae [xU2S )Z.ThenYaR[Dzn ݥL-:2*( ;R-:2);㍍Ud !6#wK*(`T*8&<[Dz1 9T:YaɦLHMNenadUUAntoniGc&NevenBalenoviGc2(K6q6COptimalUUdesignforplatesandrelaxationN퍑SProMofofthistheoremusessimilarargumen!tsastheonedemonstratedabo!ve.Howeverit 33hea!vily!reliesontheuseofoscillatingtestfunctions[T]!andsomeexplicitconstructions,@whichmak!esfitquitetechnical;thereforewechosetoomitit.33TheıAuthorswishtothankProfessorLucTeartarforsho!wingthemsomeofhisunpublishednotes.!^[ReferencesyS[AB]N.SAn!toni"Dc,dN.Balenovi"Dc:iZHomogenisationofthefourth{orderellipticequationsS,dinprepara-tionu[B]J.M.Ball: ZAv!ersionofthefundamentaltheoremforYeoungmeasuresS,1inY': 3 cmti10YPDE'sdandContin-uumMopdelsofPhaseT)ransitionsS,bM.QRascle,D.SerreandM.SlemroMd(eds.),LectureNotesinfPh!ysics,Veol.344,Springer1989,pp.207{215y[BV]E. Bonnetier,6M.Veogelius:*ZRelaxationofacompliancefunctionalforaplateoptimisationproblemS,MinFYApplicpations.3ofMultipleScalinginMechanicsS,MPe.FG.Ciarlet,E.Sfanc!hez{Palencia(eds.),fMasson1987,pp.31{53c[KV]R.-kV.Kohn,EM.Veogelius:_ZThinplateswithvdDaryingthic!kness,andtheirrelationtostructuraloptimisationS,Murat,dL.Teartar:PZH-con!vergenceS,in>YT)opicsvinthemathematicpalmodelFlingofcompositematerialsS,fA.CherkdDaev,R.Kohn(eds.),Birkhfauser,1997,pp.21{43[T]L.RTeartar:ZHomogenization,bCompMensatedCompactness,H-measuresS,lecturenotesinprepa-ration[;jZK!ON]V.MV.L;jZik!ov,_TS.MM.Kozlo!v,O.A.Oleinik,KhaT'enNgoan:rZAv!eragingandTGZ-convergenceofdi eren!tial7opMeratorsS,MYR\ussianMath.Surveys[34S(1979)69{147[YUsppehimat.naukes[34S(1979)65{133]MNenadUUAntoniGc&NevenBalenoviGc2(K7;D{Nff cmbx12kN cmbx12dg cmmi12a 3 msam10`%n 3 eufm10^kAH 3 cmssbx10]m#R 3 cmss10\ 3 cmmi10SK`y 3 cmr10Mm#R cmss10K"V cmbx10CK`y cmr10:q[ cmsl93o cmr9+2@cmbx8%K cmsy8$2cmmi8#|{Ycmr8 q% cmsy6 ;cmmi6 Aacmr60