Topics in Regularity and Qualitative Properties of

TopicsinRegularityandQualitativePropertiesofSolutionsofNonlinearEllipticEquationsXavierCabreDepartamentdeMatematicaAplicada1Universita[e-mail]:theKrylov-SafonovHarnackinequality4.Maximumprinciplein\narrowdomains5.Positivesolutionsinsomeunboundeddomains6.Fullynonlinearequations:denitionsandexamples7.C1;estimateforclassicalsolutionsofF(D2u)=08.ViscositysolutionsandJensen’sapproximatesolutionsReferencesIntroductionInthesenoteswedescribetheAlexandro-Bakelman-PucciestimateandtheKrylov-SafonovHarnackinequalityforsolutionsofLu=f(x),whereLisasecondorderuniformlyellipticoperatorinnondivergenceformLu=aij(x)@iju+bi(x)@iu+c(x)u;withboundedmeasurablecoecientsinadomainofRn.TheseinequalitiesdonotrequireanyregularityofthecoecientsofL,andthismakesthempowerfultoolsinthestudyofsecondordernonlinearellipticequations.Itisthepurposeofthesenotestopresentseveraloftheirapplicationsinthiseld.ThersttopicisthestudyofthemaximumprinciplefortheoperatorLanditsapplicationstosymmetrypropertiesofpositivesolutionsofsemilinearproblems(u+f(u)=0inu=0on@:Usingthemovingplanesmethod,weprovethesymmetryresultofGidas,NiandNirenberg[16],intheimprovedversionofBerestyckiandNirenberg[7]whichusesthemaximumprincipleindomainsofsmallmeasure.In[16,7]thesamemethodisusedtoprovesymmetryresultsforsomefullynonlinearellipticequationsF(x;u;Du;D2u)=0:Next,wepresentashortproofofseveralestimatesandmaximumprinciples(takenfrom[8]and[9])forsolutionsin\narrowdomains.WediscussalsorecentworkofBerestycki,CaarelliandNirenberg[5]onqualitativepropertiesofpositivesolutionsinsomeunboundeddomainsofcylindricaltype.Thesecondtopicthatwetreatistheregularitytheoryforsolutionsoffullynonlinearellipticequations.Ourpresentationisonlyarstandshortintroductiontothistopic;see[12,17]formoredetailedexpositions.Westartgivingimportantexamplesoffullynonlinearellipticequations:Bellmanequationsinstochasticcontroltheory,Isaacsequationsindierentialgames,theMonge-Ampereequation,andtheequationofprescribedGausscurvature.WeproveaC1;estimateforclassicalsolutionsoffullynonlinearequationsoftheformF(D2u)=0:ThemaintoolemployedhereistheCregularityforsolutionsoflinearequationsLu=0withboundedmeasurablecoecients,whichisaconsequenceoftheKrylov-SafonovHarnackinequality.1Next,weintroducethenotionofviscositysolutionofafullynonlinearellipticequa-tionandwegivethebasicpropertiesofthisclassofsolutions.Finally,wepresentJensen’sapproximatesolutions[19].Theyconstituteakeytoolwhenprovingunique-nessandregularityforviscositysolutions|atopicthatweomithere.WealsoomittheimportantC2;regularitytheoryofEvansandKrylovforconvexfullynonlinearequations(see[12,17]).Theresultspresentedinthesenotesareasamplefromthevastliteratureonthemaximumprinciple,symmetrypropertiesandregularitytheoryforfullynonlinearequations.Someofthemarefundamentalresultsinthesetheories.Othershavebeenselectedtoillustratethemaintechniquesusedintheseeldsofresearch.ThesenotesarebasedoncoursesgivenattheEcoleDoctoraledeMathematiquesetdeMecaniquedel’UniversitePaulSabatier(Toulouse),andattheCIMPAInterna-tionalSchoolinPDE’s(Temuco,Chile)organizedbytheUniversidaddeChile.Theauthorwouldliketothanktheseinstitutionsfortheirinvitations.HealsothanksIanSchindlerforhisvaluablehelptypingandcorrectingtherstdraftofthesenotes.1TheAlexandro-Bakelman-PucciestimateThroughoutthesenotes,LwilldenoteanellipticoperatorinadomainRn,oftheformLu=aij(x)@iju+bi(x)@iu+c(x)u(wheresummationoverrepeatedindicesisunderstood).WeassumethatLisuni-formlyellipticandthatithasboundedmeasurablecoecients.Thatis,wesupposethatthereexistconstants0c0C0,b0and~b0suchthatc0jj2aij(x)ijC0jj2(Pbi(x)2)1=2bjc(x)j~b:forallx2and2Rn.Hence,thematrixA(x)=[aij(x)](whichisassumedtobesymmetric)hasallitseigenvaluesintheinterval[c0;C0].Foragivenfunctionf:!R,weconsiderthelinearequationLu=f(x).Itiscalledasecondorderuniformlyellipticequationinnondivergenceformwithboundedmeasurablecoecients.UndernofurtherassumptionsonthecoecientsofL,thefollowingbasicestimate(whichwecallAPBestimate)wasprovenindependentlybyAlexandro,BakelmanandPucciinthesixties[1,2,3,25].2Theorem1.1.(Alexandro,Bakelman,Pucci)Weassumethatisaboundeddo-mainofRnandthatc0in.Letdbeaconstantsuchthatdiam()d.Letu2W2;nloc()andf2Ln()satisfyLufinandlimsupx!@u(x)0.ThensupuCdiam()kfkLn();whereC=C(n;c0;bd)isaconstantdependingonlyonn,c0andbd.HereW2;nloc()denotestheSobolevspaceoffunctionsthat,togetherwiththeirsecondderivatives,belongtoLnloc().RecallthatnisthedimensionofthespaceandthatW2;nloc()C()|thespaceofcontinuousfunctionsin.Ifu2C()thentheconditionlimsupx!@u(x)0meanssimplythatu0on@.WhenLufwesaythatuisasubsolutionoftheequationLu=f.IfLufinbuttheassumptionlimsupx!@u(x)0isnotsatised,anestimateforsupumaybeobtainedbyapplyingTheorem1.1toulimsupx!@u+(x).Wehavesupulimsupx!@u+(x)+Cdiam()kfkLn();whereu+=max(u;0)denotesthepositivep