Local behavior of solutions of quasi-linear equations

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1、LOCALBEHAVIOROFSOLUTIONSOFQUASI-LINEAREQUATIONSBYJAMESSERRINUniversityofMinnesota,U.S.A.Thispaperdealswiththelocalbehaviorofsolutionsofquasi-linearpartialdif-ferentialequationsofsecondorderinn/>2independentvariables.~reshallbeconcernedspecificallywiththeapriorimajori

2、zationofsolutions,thenatureofremovablesin-gularities,andthebehaviorofapositivesolutionintheneighborhoodofanisolatedsingularity.CorrespondingresultsareforthemostpartwellknownforthecaseoftheLaplaceequation;roughlyspeaking,ourworkconstitutesanextensionoftheseresultstoaw

3、ideclassofnon-linearequations.Throughoutthepaperweareconcernedwithrealquasi-linearequationsofthegeneralformdiv.,4(x,u,uz)=~(x,u,ux).(1)Here~isagivenvectorfunctionofthevariablesx,u,ux,~isagivenscalarfunc-tionofthesamevariables,andu~=(Ou/Ox1.....Ou/Oxn)denotesthegradie

4、ntofthedependentvariableu=u(x),wherex=(xl,...,xn).Thestructureof(1)isdeterminedbythefunctions.,4(x,u,p)andB(x,u,p).Weassumethattheyaredefinedforallpointsxinsomeconnectedopenset(domain)~oftheEuclideannumberspaceEn,andforallvaluesofuandp.Furthermore,theyaretosatisfyine

5、qualitiesoftheformIAI1isafixedexponent,aisapositiveconstant,andthecoefficientsbthrough9aremeasurablefunctionsofx,containedincertaindefiniteLebesgueclassesover(seeChapterI).248JAM~.SSERRINThegeneralityoftheseassumptionsnat

6、urallyrequiresthatequation(1)beinter-pretedinaweaksense.LetDbeasubdomainof~,andletubeafunctionhavingstrongderivativesuxwhicharelocallyofclassL~overD.Thenuwillbecalledaweaksolution(orsimplyasolution)of(1)inDiff(rA+~)dz=0(3)foranycontinuouslydifferentiablefunction~b=rw

7、ithcompactsupportinD.Ob-viouslyanyfunctionwhichsatisfies(1)intheclassicalsensewouldbeasolutioninthesensejustdefined,thoughofcoursenotconversely.Beforeturningtoadescriptionofresults,itisworthnotingthattheabovestructureincludeslinearellipticequations,wherea=2,andalsoth

8、eEulerequationsofawideclassofregularaswellasnon-regularvariationalproblems.Thereadermayconsultreference[22],wherethisobservationism