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1、Vol.57,No.3DUKEMATHEMATICALJOURNAL(C)December1988ONPOSITIVESOLUTIONSOFSECOND-ORDERELLIPTICEQUATIONS,STABILITYRESULTS,ANDCLASSIFICATIONYEHUDAPINCHOVER0.Introduction.LetPbeasecond-orderuniformlyellipticoperatordefinedinadomainfl__cRn.Let'v(f)betheconvexco
2、neof'allpositivesolutionsoftheequationPu0inf.(0.1)IfoneassumesthatPorf]satisfiessomeveryspecialproperties,e.g.,thatPandf]areinvariantunderrotationsortranslations,thenonecangetanexplicitdescriptionofthestructureof'e(fl)andalsodelicateperturbationresults(
3、see,forexample,[2],[4],[8],[9],[11]).Inthispaperouraimisdifferent.Wewishtodiscusssomegeneralpropertiesofe(f)forgeneralPandf]andtoobtainsomeinformationonthestructureofe(f).Thepaperisdividedintofoursections.Insection1weshallgivesomebasicdefinitions,fixnot
4、ations,andrecallbrieflysomeknownresults.Insection2weshallcomparethestructureof'e(f),i--1,2,whereP2isa"small"perturbationofPx.Itturnsoutthatinthegeneralcasea"small"perturbationisachangeofthecoefficientsofPxinacompactsetinf].Weshallseeinsection3thatunders
5、omeassumptionstheanalogueforasmallperturbationforthecasef]RnisaperturbationoftheformP2P1+W,whereWLI(R).Insection4weshallstudythequestionofstabilityandinstabilityof'e(f])byinvestigatingthebehaviorof,+tw(f)whenRisvariedandWisafixedfunction.Weshallconfineo
6、urselvestoclassicalsolutions.Theresultsarealsovalidforweakandstrongsolutions;theproofsdifferonlyinminordetailsfromtheproofsfortheclassicalsolutions.Someoftheresultsinthispaperwereannouncedfirstin[9].IwouldliketoremarkthatsimilarresultswereobtainedbyM.Mu
7、rataforSchriSdingeroperators([8];seealso[3]).Acknowledgements.Thepaperisbasedonpartoftheauthor'sPh.D.thesis,completedin1986attheHebrewUniversityofJerusalem[10].Iwishtoexpressmydeepgratitudetomythesisadvisor,ProfessorShmuelAgmon,fortheencouragement,suppo
8、rt,andhelphegaveme.ReceivedOctober22,1987.955956YEHUDAPINCHOVER1.Preliminaries.Weshallconsiderasecond-orderellipticoperatorPactingonfunctionsuinadomainR",n>2.WeassumethatPisauniformlyellipticoperatoroftheformPu=aij(x)8i3ju+bi(x)3
