The comparsion principle for viscosity solutions o

arXiv:math/0309078v1[math.AP]4Sep2003ThecomparsionprincipleforviscositysolutionsoffullynonlinearsubellipticequationsinCarnotgroupsChangyouWangDepartmentofMathematics,UniversityofKentuckyLexington,KY40506Abstract.ForanyCarnotgroupGandaboundeddomainΩ⊂G,weprovethatviscositysolutionsinC(¯Ω)ofthefullynonlinearsubellipticequationF(u,∇hu,∇2hu)=0areuniquewhenF∈C(R×Rm×S(m))satisfies(i)Fisdegeneratesubellipticanddecreasinginuor(ii)Fisuniformlysubellipticandnonincreasinginu.ThisextendsJensen’suniquenesstheoremfromtheEuclideanspacetothesub-RiemanniansettingoftheCarnotgroup.§1.IntroductionThenotionofviscositysolutionsoffullynonlinear2ndorderdegenerateellipticequa-tion:F(x,u(x),∇u(x),∇2u(x))=0,inRn,(1.1)wasdevelopedbyCrandall-Lions[CL]andEvans[E1,2]in1980’s.Thisidea,togetherwithJensen’scelebrateduniquenesstheorem[J1],providesaverysatisfactorytheoryonexistence,uniqueness,andcompactnesstheoremofweaksolutionsof(1.1).Thetheoryofviscositysolutionshasbeenverypowerfulinmanyapplications,andwerefertotheuser’sguide[CIL]byCrandall-Ishii-Lionsformanysuchapplications.Inrecentyearstherehasbeenanexplosionofinterestinthestudyofanalysisonsub-Riemannian,orCarnot-Carath´edoryspaces.ThecorrespondingdevelopmentsinthetheoryofpartialdifferentialequationsofsubelliptictypehavepromptedpeopletoconsiderfullynonliearequationsinCarnotgroups.Forexamples,motivatedbytheveryimportantworkofJensen[J2]onabsoluteminimizingLipschitzextensions(orALMEs,anotionfirstintroducedbyAronsson[A])andviscositysolutionstothe∞-laplacianequationintheEuclideanspace,Bieske[B],Bieske-Capogna[BC],andWang[W1]havestudiedabsoluteminimizinghorizontalLipschitzextensionsandviscositysolutionstothe∞-sublaplacianequationonCarnotgroups.Inparticular,thenotionofviscositysolutionshasbeenex-tendedtofullynonlinearsubellipticequation(see[B])andtheuniquenessofviscositysolutionof∞-sublaplacianeqautiononanyCarnotgroupwasestablishedbyWang[W1].Itiswell-known(cf.themonographs[CC]byCaffarelli-Cabr´eand[G]byGutierrez)thatbothconvexityandtheMonge-Amp´ereequation:det(∇2u)=f,inRn(1.2)1haveplayedcrucialrolesinthetheoryoffullynonlinearellipticequation.Inspiredbythis,Lu-Manfredi-Stroffolini[LMS]andDanielli-Garofalo-Nhieu[DGN]haveintroducedandstudiedvariousnotionsofconvexity,suchasv-convexityandh-convexity,onCarnotgroups(seealso[BR],[W2],[JM]forsomefurtherrelatedresults).Moreover,Garofalo-Tournier[GN]andGutierrez-Montanari[GM]haveinitiatedthestudyofMonge-Amp´eremeasuresandmaximumprincipleofconvexfunctionsonHeisenberggroups.Inthispaper,weareinterestedinthecomparisonprincipleforviscositysolutionsto2ndordersubellipticequationwhichiseitheruniformlysubelliptic,nonincreasingordegeneratesubelliptic,decreasinginthesub-RiemanniansettingoftheCarnotgroup.Inthisaspect,weareabletoextendJensen’suniquenesstheoremfromtheEuclieanspacetoanyCarnotgroup.Inordertodescribeourresult,wefirstrecallthebasicpropertiesofCarnotgroups.AsimplyconnectedLiegroupGiscalledaCarnotgroupofstepr≥1,ifitsLiealgebragadmitsavectorspacedecompositioninrlayersg=V1+V2+···+Vrsuchthat(i)gisstratified,i.e.,[V1,Vj]=Vj+1,j=1,···,r−1,and(ii)gisr-nilpotent,i.e.[Vj,Vr]=0,j=1,···,r.WecallV1thehorizontallayerandVj,j=2,···,rtheverticallayers.Wechooseaninnerproducth·,·iongsuchthatV′jsaremutuallyorthogonalfor1≤j≤r.Let{Xj,1,···,Xj,mj}denoteafixedorthonormalbasisofVjfor1≤j≤r,wheremj=dim(Vj)isthedimensionofVj.Fromnowon,wealsodenotem=dim(V1)asthedimensionofthehorizontallayerandsetXi=X1,ifor1≤i≤m.Itiswell-known(see[FS])thattheexponentialmapexp:g≡Rn→GisaglobaldiffeomorphismandyieldsanexponentialcoordinatesystemonG,withn=Pri=imithetopologicaldimensionofG.Moreprecisely,anyp∈Ghasacoordinate((p1,···,pm),(p2,1,···,p2,m2),···,(pr,1,···,pr,mr))suchthatp=exp(ξ1(p)+···ξr(p)),withξ1(p)=mXl=1plXl,ξi(p)=miXj=1pi,jXi,j,2≤i≤r.Theexponentialmapcaninduceahomogeneouspseudo-normNGonGinthefollowingway(see[FS]).NG(p):=(rXi=1|ξi(p)|2r!i)12r!,ifp=exp(ξ1(p)+···ξr(p)),(1.3)where|ξ1(p)|=(Pml=1p2l)12,and|ξi(p)|=(Pmij=1p2i,j)12(2≤i≤r).Moreover,NGyieldsapseudo-distanceonGasfollows.dG(p,q):=NG(p−1·q),∀p,q∈G,(1.4)where·isthegroupmultiplicationofGandp−1istheinverseofp.ItiseasytoseethatdGsatisfiestheinvariancepropertydG(z·x,z·y)=dG(x,y),∀x,y,z∈G,(1.5)2andisofhomogeneousofdegreeone,i.e.dG(δλ(p),δλ(q))=λdG(p,q),∀λ0,∀p,q∈G(1.6)whereδλ(p)=λξ1(p)+Pri=2λiξi(p)isthenon-isotropicdilationsonG.Throughoutthispaper,wefixsomenotations.Forl≥1,denoteS(l)asthesetofl×lsymmetricmatrices.ForM,N∈S(m),wesayM≥Nif(M−N)∈S(m)isapositivesemidefinitematrix,andlettrace(M)denotethetraceofMforM∈S(m).Foru:G→R,let∇u,∇2udenotetheEuclideangradient,hessianofurespectively,and∇hu:=(X1u,···,Xmu),∇2hu:=(XiXj+XjXi2u)1≤i,j≤mdenotethehorizontalgradient,horizontalhessianofurespectively.ForagivendomainΩ⊂G,denoteC(Ω)asthesetofcontinuousfunctionsonΩ,C2(Ω)={u∈C(Ω):∇u,∇2u∈C(Ω)},andΓ2(Ω)={u∈C(Ω):∇hu,∇2hu∈C(Ω)}.Afullynonlinearpartialhorizontal-differentialoperatorF[·]onΩisdefinedbyF[φ](x)=F(φ(x),∇hφ(x),∇2hφ(x)),∀x∈Ω,∀φ∈Γ2(Ω),(1.7)whereF∈C(R×Rm×S(m)).Wenowgivethedefinitionofsubellipticityandnonde-creasingpropertyofF.Definition1.1.TheoperatorF[·]isdegeneratesubelliptici