; TeX output 2001.06.13:1514 hGS AK`y UU cmr10AAnnalesAcademiScienĽtiarumFN8ennic MathematicaVN8olumen26,2001,509{518"27"V cmbx10HIGHERORDERVwxARIAٚTIONALINEQUALITIESW dWITHNON-STٚANDARDGROWTHCONDITIONS4INDIMENSIONTWO:PLAٚTESWITHOBSTACLES `@"V cmbx10Mic9haelTBildhauerandMartinF uchsVGAUnivĽersitUatdesSaarlandes,FN8achrichtung6.1Mathematik qPĽostfach151150,D-66041Saarbr;CucĽken,Germany;bibi@math.uni-sb.de,fuchs@math.uni-sb.de L"V UU cmbx10LAbstract. AFN8or;adomain; 7pH!", UU cmsy10HLRBٓR U cmr7B2AwĽe;considerthesecondorderv{ariationalproblemof minimizing:Db> UU cmmi10DJ /A(Dw9gA)=IyKu UU cmex10KRB .Df A(HrB2_DwA)cDdx:Aamong:functionsDw9gA:a H!LR:Awithzerotraceresp;BectingasideconditionLoftheformLDwHA on .HereDfmAisLasmo;BothconĽvexLintegrandwithnon-standardgroĽwth,a typicalexampleisgivenby Df A(HrB2_Dw9gA)=HjrB2Dw9gHjcAlnUT(1v+HjrB2Dw9gHjA) .tWN8e proĽvethat|undersuitableassumptionson |theuniqueminimizerisofclassDC B1E0er U cmmi7E; A( )foranĽyDXy<PA1 .XOurresultsproĽvideakindofinterp;BolationbetĽweenelasticandplasticplateswithobstacles. [mX1.pIn9troQductionTandmainresult K`y cmr10Let\ denote\abGounded,star-shaped\Lipschitzdomainin\R^ٓR cmr72 GϫandsuppGosewearegivenanb> cmmi10N-functionAhavingthe2|s-propGerty*,#precisely(see,e.g.[A]yfordetails)UUthefunctionUUA:UP[0;!", cmsy101)![0;1)UUsatises FQAUUiscontinuous,UUstrictlyincreasingandconvex{;(N1)܍HlimIk 0er cmmi7tO! cmsy7#0<$YKeA(t)YKew fe 㑟 (֍tp)A=0;4rlim t!1<$+A(t)+w fe 㑟 (֍tB]=+1;(N2)cFQthereUUexistxUkP;UPt0C0:mA(2t)kA(t) forUUall11ͱtt0|s:(N3)TheSfunctionSAgeneratestheOrliczspaceLA( )equippGedwiththeLuxem-burgUUnormqYnkuk:LmO \ cmmi5A>( )g#:=infƟ^u cmex10qDZlx>0:cZUR y 5QA^<$1w fe (֍ l۷juj^ dx1^;ՍtheOrlicz{SobGolevspaceW^clbA( )isdenedinastandardway(seeagain[A]),nally*,UUweletx䍑g6fWclAv ( ):=mclosureUUof2ꪱC 1፮0 0( )UUin WclA( ): [F*orloGcalspacesweusesymbGolslikex䍑W^clvA;lo7c 3( ) ,αL1ɍpvlo7c X( )etc.HOSuppGosefurtherthatweUUaregivenafunctionUU 2W^c3l2( )( C ^1;q(} fe 8 ))UUwhichsatises :j@n9 <0;maxs,ޟH fe g )ȫ >0 ff xsUUA2000MathematicsSubvjectClassication:%Primary49N60,74K20,35J85. * 510Y!p0J cmsl10MichaelUUBildhauerandMartinF*uchshGandUUlet j(K:=`n qıv"2x䍑qëWc2፴Ao( ):v UUa.e.on#< `o:oIt;iseasytoseethat;Kcontains;afunction; 0NofclassC^ 1l0 0( ) :0:let ^Z cmr5+j:=[ 0]and@qchoGose@qȷ2NC^ 1l0 0( )such@qthatȷN1on ^+ and0ȷ1on .3Then 0C:=maxbxN0;max[ fe g b *hasUUthedesiredpropGerties.Next weformulatethehypGothesesimposedontheintegrand: 7f:UPR^22ڷ![0;1)UUisUUofclassC ^2 satisfyingHlmAc1|sb QʱAb WjuǷjb7 81b %f(uǫ)c2|sb QʱAb Wjjb7+81b W;(1:1)%\5b W18+juǷj2|sb ʟߺ =2j[ٷj2 %DG2Ðf(uǫ)([;);(1:2)#ݍ /jDG2Ðf(uǫ)j %<+1;(1:3) xjDG2Ðf(uǫ)jjj2 %c3|sb Qʱf(uǫ)8+1b W;(1:4) xhAb .;jDGf(uǫ)jb %c4|sb QʱAb WjuǷjb7+81b (1:5)forsallsuǫ, :"2R^22.R|Herec1|s,c2,c3,c4,sanddenotespGositiveconstants, :sis someparameterin[0;2) , andA^istheY*oungtransformofܱA.RF*rom(1.3)weseethat fӏis ofsubGquadraticgrowth,ګi.e.limsup#\tqƺj=j!1>f(uǫ)=jj^2R<x߫+1 ,ګ(1.4) istheso-calledbalancingconditionbGeingofimportancealsointhepapers[FO],[FM]and#[BFM].Asshownforexamplein[FO] wecantake#f(uǫ):=jjln (1}+jj)#or#itsiterated!"version!"flȫ(uǫ):=jj\q'!~fl()!"with\qHC~f1 ()=lno(1z+jj) ,\qR~+fl `+1B()=lnob ƫ1z+\q~fl Ǚ()bW. aButalsopGowergrowth(1+juǷj^2|s)^p=2 ,14