数学杂志Vol.43(2023)J.ofMath.(PRC)No.6THENEUMANNPROBLEMFORASPECIALLAGRANGIANTYPEEQUATIONWITHSUPERCRITICALPHASEJIAHao-hao,XUWen-zhao(SchoolofMathematicsandStatistics,NingboUniversity,Ningbo315211,China)Abstract:Inthispaper,weexploretheNeumannproblemforspeciallagrangiantypeequa-tionswithsupercriticalphaseinRr.WeshowtheglobalC2aprioriestimatesofthesolutionandestablishtheexistenceofclassicalsolutionsbythemethodsofcontinuity.Keywords:Neumannproblem;specialLagrangiantypeequation;supercriticalphase2010MRSubjectClassification:35J60;35B45Documentcode:A1IntroductionandmainresultsWeconsidertheNeumannproblemofaspecialLagrangianequationwherearctan[△uln-D?u}=:arctanni+arctann2+...+arctannn.Denoten:=(ni,n2,::,nn)whicharetheeigenvaluesofthematrixuln-D2uin[1]withwhere入=(Ai,2,*.:,An)aretheeigenvaluesoftheHessianmatrixD?u.Here(α)isusuallystudiedunderthreedifferenttypesoftwoboundaryvalueconditions:thephase,thecriticalphase,supereriticalphase.Morepreisely,(a)e(-,),e=Pr,pr(a)nhispapr,weconiderhspeiaagrangianequation(.)with2supercriticalphase,thatisthethirdtype.Thefirstboundaryvalueproblem(Dirichletproblem)forellipticpartialdifferentialequationshasbeenintensivelystudiedmanyyears.FortheLaplaceequation,resultscanbefoundinGilbarg-Trudinger[2].TheDirichletproblemforMonge-Ampereequations*Receiveddate:2022-09-04Foundationitem:SupportedbyNationalNaturalScienceFoundationofChina(12171260).Biography:JiaHaohao(1995-),female,bornatHandan,Hebei,postgraduate,majorinpartialdifferentialequations.E-mail:2101400103@nbu.edu.cnArticleID:0255-7797(2023)06-0471-16arctan[△uIn-D²u)=O(c),in2CR",n=Z入,Vi=1,2,..,n,kiAccepteddate:2022-12-05(1.1)2，22472wasinvestigatedinCaffarelli-Nirenberg-Spruck[3]andKrylov[4].Theyshowedtheglobalregularityofsolutions.Caffarelli-Nirenberg-Spruck[5]studiedtheexistenceofadmissiblesolutionsandtheglobalregularityofk-Hessianequations.TheHessianquotientequationswhichhavedifferentstructureconditionswerestudiedinTrudinger[6].Tothebestofmyknowledge,thespecialLagrangianequationJournal...
