Regularity theory for fully nonlinear integro-diff

arXiv:0709.4681v3[math.AP]26Apr2008Regularitytheoryforfullynonlinearintegro-differentialequationsLuisCaffarelliandLuisSilvestreApril26,2008AbstractWeconsidernonlinearintegro-differentialequations,liketheonesthatarisefromstochasticcontrolproblemswithpurelyjumpL`evyprocesses.WeobtainanonlocalversionoftheABPestimate,Harnackinequality,andinteriorC1,αregularityforgeneralfullynonlinearintegro-differentialequations.Ourestimatesremainuniformasthedegreeoftheequationapproachestwo,sotheycanbeseenasanaturalextensionoftheregularitytheoryforellipticpartialdifferentialequations.1IntroductionIntegro-differentialequationsappearnaturallywhenstudyingdiscontinuousstochasticprocesses.Thegeneratorofann-dimensionalL`evyprocessisgivenbyanoperatorwiththegeneralformLu(x)=Xijaij∂iju+Xibi∂iu+ZRn(u(x+y)−u(x)−∇u(x)·y)χB1(y)dμ(y).(1.1)Thefirsttermcorrespondstothediffusion,thesecondtothedrift,andthethirdtothejumppart.Inthispaperwefocusontheequationsthatweobtainwhenweconsiderpurelyjumpprocesses;processeswithoutdiffusionordriftpart.TheoperatorshavethegeneralformLu(x)=ZRn(u(x+y)−u(x)−∇u(x)·yχB1(y))dμ(y).(1.2)whereμisameasuresuchthatRRn|y|21+|y|2dμ(y)+∞.ThevalueofLu(x)iswelldefinedaslongasuisboundedinRnandC1,1atx.Theseconceptswillbemademorepreciselater.TheoperatorLdescribedaboveisalinearintegro-differentialoperator.Inthispaperwewanttoobtainresultsfornonlinearequations.Weobtainthiskindofequationsinstochasticcontrolproblems[11].Ifinastochasticgameaplayerisallowedtochoosefromdifferentstrategiesateverystepinordertomaximizetheexpectedvalueofsomefunctionatthefirstexitpointofadomain,aconvexnonlinearequationemergesIu(x)=supαLαu(x)(1.3)Inacompetitivegamewithtwoormoreplayers,morecomplicatedequationsappear.WecanobtainequationsofthetypeIu(x)=infβsupαLαβu(x)(1.4)1Thedifferencebetween(1.4)and(1.3)isconvexity.Alternatively,alsoanoperatorlikeIu(x)=supαinfβLαβu(x)canbeconsidered.AcharacteristicpropertyoftheseoperatorsisthatinfαβLαβv(x)≤I(u+v)(x)−Iu(x)≤supαβLαβv(x)(1.5)AmoregeneralandbetterdescriptionofthenonlinearoperatorswewanttodealwithistheoperatorsIforwhich(1.5)holdsforsomefamilyoflinearintegro-differentialoperatorsLαβ.TheideaisthatanestimateonI(u+v)−Iubyasuitableextremaloperatorcanbeareplacementfortheconceptofellipticity.Indeed,ifweconsidertheextremalPuccioperators[6],M+λ,ΛandM−λ,Λ,andwehaveM−λ,Λv(x)≤I(u+v)−Iu≤M+λ,Λv(x),thenitiseasytoseethatImustbeanellipticsecondorderdifferentialoperator.Ifinsteadwecomparewithsuitablenonlocalextremaloperators,wewillhaveaconceptofellipticityfornonlocalequations.Wewillgiveaprecisedefinitioninsection3(Definition3.1).WenowexplainthenaturalDirichletproblemforanonlocaloperator.LetΩbeanopendomaininRn.WearegivenafunctiongdefinedinRn\Ω,whichistheboundarycondition.WelookforafunctionusuchthatIu(x)=0foreveryx∈Ωu(x)=g(x)forx∈Rn\ΩNoticethattheboundaryconditionisgiveninthewholecomplementofΩandnotonly∂Ω.ThisisbecauseofthenonlocalcharacteroftheoperatorI.Fromthestochasticpointofview,itcorrespondstothefactthatadiscontinuousL`evyprocesscanexitthedomainΩforthefirsttimejumpingtoanypointinRn\Ω.InthispaperwewillfocusmainlyintheregularitypropertiesofsolutionstoanequationIu=0.Wewillbrieflypresentaverygeneralcomparisonprinciplefromwhichexistenceofsolutionscanbeobtainedinsmoothdomains.Inordertoobtainregularityresults,wemustassumesomenicebehaviorofthemeasuresμ.Basically,ourassumptionisthattheyaresymmetric,absolutelycontinuousandnottoodegenerate.Tofixideas,wecanthinkofintegro-differentialoperatorswithakernelcomparablewiththerespectivekernelofthefractionallaplacian−(−△)σ/2.Inthisrespect,thetheorywedevelopcanbeunderstoodasatheoryofviscositysolutionsforfullynonlinearequationsoffractionalorder.Inthispaperwewouldliketoquicklypresentthenecessarydefinitionsandthenprovesomeregularityestimates.Ourresultsinthispaperare•Acomparisonprincipleforageneralnonlinearintegro-differentialequation.•AnonlocalversionoftheAlexandroff-Backelman-Pucciestimate.•TheHarnackinequalityforintegro-differentialequationswithkernelsthatarecomparablewiththeonesofthefractionallaplacianbutcanbeverydiscontinuous.•AH¨olderregularityresultforthesameclassofequationsastheHarnackinequality.•AC1,αregularityresultforalargeclassofnonlinearintegro-differentialequations.EventhoughtherearesomeknownresultsaboutHarnackinequalitiesandH¨olderestimatesforintegro-differentialequationswitheitheranalyticalproofs[10]orprobabilisticproofs[4],[3],[5],[12],theestimatesinallthesepreviousresultsblowupastheorderoftheequationapproaches2.Inthisway,theydonotgeneralizetoellipticdifferentialequations.Weprovideestimatesthatremainuniforminthedegreeandthereforemakethetheoryofintegro-differentialequations2andellipticdifferentialequationsappearsomewhatunified.Consequently,ourproofsaremoreinvolvedthantheonesinthebibliography.Inthispaperweonlyconsidernonlinearoperatorsthataretranslationinvariant.Thevariablecoefficientcasewillbeconsideredinfuturework.Ifuturepapers,wearealsoplanningtoaddresstheproblemoftheinteriorregularityoftheintegro-differentialHamilton-Jacobi-Bellmanequation.Thisreferstotheequationinvolvingaconvexnonlocaloperatorlike(1.3).Inthatcaseweobtaina