非线性偏微分方程cnskii

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1、TheDirichletProblemforNonlinearSecond-OrderEllipticEquations.11.ComplexMonge-Ampere,andUniformlyElliptic,EquationsL.CAFFARELLIUniversityofChicagoJ.J.KOHNPrincetonUniversityL.NIRENBERGCourantInstituteANDJ.SPRUCKUniversityofMassachusettsatAmherstIntroductionThisisthesecondpaperinaseries*devotedtothe

2、Dirichletproblemforsecond-ordernonlinearellipticequationsforarealfunctioninaboundeddomain0inR"withsmoothboundaryail.WetreattheproblemF(X,U,DU,D~U)=0in0,u=gconaR.ThefunctionFisassumedtobesmoothforxEaandinallthearguments,whereverconsidered(seehowevertheRemarkonpage2431,andtheequationisassumedtobeell

3、iptic,i.e,,ZPgi5j>0for6=(t,,*.*,f)#0;here*Part111willappearinActaMathematica.CommunicationsonPureandAppliedMathematics,Vol.XXXVIII,209-252(1985)01985JohnWiley&Sons,Inc.CCC0010-3640/85/020209-44$04.00210L.CAFFARELLI,J.J.KOHN,L.NIRENBERG,ANDJ.SPRUCKInaddition,asintheworkofL.C.Evans[12],[13],wealways

4、assumethatforthevaluesoftheargumentsunderconsideration,(3)Fisaconcavefunctionofthesecondderivatives{u,}.(Inplaceof(3)onemayconsiderFconvexin{u,}byreplacingFanduby-Fand-U.)Inthefirstpaper[8]weconsideredMonge-Ampereequationsoftheformdet(u,)=$(x,u,Vu);herelog(det(u,))satisfies(3)forstrictlyconvexfunc

5、tionsu.Inthatpaperwealsoexplainedtheuseofthecontinuitymethodforattacking(l),(1)'andtheuseofaprioriestimatesfortheC2+.(n)normofu,0

6、reetopics,presentedinthreesections.InSection1weextendtheC2aprioriestimateof[8]toellipticcomplexMonge-Ampereequationsoftheformdet(u,,,)=$(z,u,Bu)>0inn,u=cponan.HereRisastronglypseudo-convexdomaininC*,-2=(z,,-..,z,)En,zj=xj+iy,,u-=d..u='z(axk+iayk)u9UZI=a,u=&(a,,-ia,,)u,ZkZkuZ,?,='9etc.Weseekastrict

7、lypluri-subharmonicsolutionu,i.e.,asolutionforwhichthecomplexHessianmatrix{u~,~~}ispositivedefinite.InSection1wederivetheaprioriestimatefortheC2(B)norm(6)lulzIC'.Inadditionweestablishestimatesforcertainclassesoff