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1、Math.Z.(2012)272:405427DOI10.1007/s00209-011-0940-5MathematischeZeitschriftPositivesolutionsofSchrödingerequationsandfineregularityofboundarypointsAlanoAnconaReceived:31March2011/Accepted:2August2011/Publishedonline:30September2011©Springer-Verlag2011AbstractGivenaLipschi
2、tzdomaininRNandanonnegativepotentialVinsuchthatV(x)d(x,∂)2isboundedwestudythefineregularityofboundarypointswithrespecttotheSchrödingeroperatorLV:=−Vin.Usingpotentialtheoreticmethods,severalconditionsareshowntobeequivalenttothefineregularityofz∈∂.Themainresultisasimpl
3、e(explicitifissmooth)necessaryandsufficientconditioninvolvingthesizeofVforz∈∂tobefinelyregular.Anintermediateresultconsistsinamajorizationof
4、u
5、2dxforupositiveharmonicinandA⊂.ConditionsforalmosteverywhereAd(.,∂)regularityinasubsetAof∂arealsogivenaswellasanextensionof
6、themainresultstoanotionoffineL1
7、L0-regularity,ifLj=L−Vj,V0,V1beingtwopotentials,withV0≤V1andLasecondorderellipticoperator.MathematicsSubjectClassification(2000)31C35·31C25·35J15·35C151IntroductionandmainresultsThispaperstemsfromaquestionraisedbyMosheMarcusandLaurentVéronse
8、veralyearsagoanditsanswer[7](seetheappendixin[32]andthecommentsafterTheorem4.2below).AnotherindependentandrecentquestionofMosheMarcushasalsomotivatedtheapproachfollowedinSect.3.SeeProposition2.6.BothquestionsdealtwithpositivesolutionsofaSchrödingerequationu−Vu=0inaLipsc
9、hitzdomainwithVinanaturalclassofnonnegativepotentialsin.Thefirstwasaboutanecessaryexplicitconditionforaboundarypointoftobefinelyregularwithrespectto−Vandisrelatedtotheworks[23,24].Thedefinitionoffineregularity[32]recalledbelowgoesbacktoanotionintroducedbyE.B.Dynkintostu
10、dytheboundaryvalues(ortraceson∂)ofpositivesolutionsofnonlinearequationsuchasu=uα,α>1inA.Ancona(B)DépartementdeMathématiques,UniversitéParis-Sud11,91405Orsay,Francee-mail:alano.ancona@math.u-psud.fr123406A.Anconawhichcase,givenasolutionu,Dynkinsdefinitioncorrespondshere
11、toV=
12、u
13、α−1(see[13,16]andthebooks[14,15]).Tostateourmainresultswefixsomenotationsandrecallbasicdefinition
