symmetrization for fractional elliptic and parabolic equations and an isoperimetric application yannick sire资料.pdf

该【symmetrization for fractional elliptic and parabolic equations and an isoperimetric application yannick sire资料 】是由【小舍儿】上传分享，beplayapp体育下载一共【26】页，该beplayapp体育下载可以免费在线阅读，需要了解更多关于【symmetrization for fractional elliptic and parabolic equations and an isoperimetric application yannick sire资料 】的内容，可以使用beplayapp体育下载的站内搜索功能，选择自己适合的beplayapp体育下载，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此beplayapp体育下载到您的设备，方便您编辑和打印。:..(2),2017,661–686ChineseAnnalsofDOI:-017-1089-2Mathematics,SeriesBcTheEditorialO?ceofCAMandSpringer-VerlagBerlinHeidelberg2017SymmetrizationforFractionalEllipticandParabolicEquationsandanIsoperimetricApplication?YannickSIRE1JuanLuisVAZQUEZ′2BrunoVOLZONE3(DedicatedtoHaimBrezis,greatmasterofanalysis,onhis70thbirthday)--calledrestrictedfractionalLaplaciande?,anoriginalproofofthecorrespondingfractionalFaber-,FractionalLaplacian,Nonlocalellipticandparabolicequations,Faber-Krahninequality2000MRSubjectClassi?cation35B45,35R11,35J61,35K551IntroductionInthispaper,wedevelopfurtherthetheoryofsymmetrizationforfractionalLaplacianoperatorsinitiatedin[25,64–65],bothintheellipticandtheparabolicsetting,byextendingittoanaturalversionofthefractionalLaplaciande?,retiveoperators,withtherecentinterestinnonlocalversionsofthedi?usionoperators,,wederiveanoriginalproofoftheFaber-Krahninequality(FKIforshort)forsuchoperatorsde?,wereviewinthisintroductionthenec-essaryinformationaboutsymmetrization,theelliptic-to-parabolictechniqueusedtogenerateManuscriptreceivedJune25,,,KriegerHall,,Baltimore,MD,21218,-mail:******@′aticas,UniversidadAut′onomadeMadrid,28049Madrid,-mail:juanluis.******@,UniversitdegliStudidiNapoli“Parthenope”,-mail:bruno.******@?ThisworkwassupportedbytheANRprojects“HAB”and“NONLOCAL”,theSpanishResearchProjectMTM2011-24696,andtheINDAM-GNAMPAProject2014“Analisiqualitativadisoluzionidiequazioniellitticheedievoluzione”(Italy).:..,′,theprecisede?nitionofthefractionalLaplacianoperatorsandtherela-?nitionsoffractionalLaplaciansonboundeddomainsWhenworkinginthewholespacedomainRN,thereareseveralequivalentde?nitionsofthefractionalLaplacianσ2operator(?Δ),0<σ<2,classicalreferencesbeing[39,53].Theinterestintheseoperatorsσ2hasalonghistoryinprobabilitysincethefractionalLaplacianoperatorsoftheform(?Δ)arein?nitesimalgeneratorsofstableL′evyprocesses(see[1,7,57]).Furthermotivationandreferencesontheliteraturearegivenforinstancein[11,64].Aparticularde?nitionthatwasconvenientforsymmetrizationpurposesde?nestheoperatorforeverygiven0<σ<2asthetraceofasuitableDirichlet-Neumannproblemviaanextendedpotentialfunctionwthatsolvesanellipticequationinanupperhalf-spaceinH+=RN×(0,∞)?RN+?arelli-Silvestreextension(see[18]).Itallowstoreducenonlocalproblemsinvolvingσ2(?Δ)tosuitablelocalproblems(actually,adegenerate-singularellipticequation),de??RN,plicatedbecausethereareσ2severaloptionsforde?ningthefractionalLaplacianoperator(?Δ).[65]onsymmetrization,wefollowedoneofhefractionalLaplacianastheDirichlet-to-Neumannmap,throughanextendedpotentialfunctionde?nedinacylinderC=Ω×(0,∞)?RN+1,aswasproposedin[17,22].(LstandsfortheLaplacian).Thissettingallowedustoderivein[65]thedesiredsymmetrizationresults,,wetakethesecondusualapproachtode?ne(?Δ),?nitionoffractionalLaplacianinRNbutaskingittoactonthenull-extensionstoRNoffunctionsu(x)de?,monformulationwithahyper-singularkernelσ2f(x)?f(y)(?Δ)f(x)=c(N,σ)N+σdy()RN|x?y|ontheconditionthatf(y)=0fory∈(see[49,10]),?,MusinaandNazarov[44]usedthenamefrac-tionalLaplacianwithNavierconditionsforthespectralversion,,wewanttoextendtooperatorL2thesymmetrizationtheorywehaddevelopedforL1in[25,64].Thishasanindependentinterestsincetherearesubtledi?erencesbetweenthetwooperators(see[10,12]).SymmetrizationSymmetrizationisaveryancientgeometricalideathatisusednowadays:..SymmetrizationforFractionalEllipticandParabolicEquations663asane?cienttoolofobtainingaprioriestimatesforthesolutionsofdi?erentpartialdi?erentialequations,-known,[34,47].TheapplicationofSchwarzsymmetrizationtoobtainingaprioriestimatesforellipticproblemswasalreadydescribedin[42,66].ThestandardellipticresultreferstothesolutionsofanequationoftheformLu=f,Lu=??i(aij?ju)i,jposedinaboundeddomainΩ?RN;thecoe?cients{aij}areassumedtobebounded,mea-surableandsatisfytheusualellipticitycondition;?nally,wetakezeroDirichletboundaryconditionsontheboundary?[54–55]leadstoparisonbetweenthesymmetrizedversion(morepreciselythesphericaldecreasingrearrangement)oftheactualsolutionoftheproblemu(x)andtheradiallysymmetricsolutionv(|x|),-parisonofconcentrations(see[2–3,58]).Thelatterconsiderstheevolutionproblemsoftheform?tu=ΔA(u),u(0)=u0,()whereAisamonotoneincreasingrealfunction,,theproblemwasposedforx∈RN,-LiggettImplicitDiscretizationtheorem(see[24])toreducetheevolutionproblemtoasequenceofnonlinearellipticproblemsoftheiterativeform?hΔA(u(tk))+u(tk)=u(tk?1),k=1,2,···,()wheretk=kh,andh>(u)=v,theresultingchainmonformhLv+B(v)=f,B=A?1.()Generaltheoryoftheseequations(see[8]),ensuresthatthesolutionmapT:f→u=B(v)isacontractioninsomeBanachspace,whichhappenstobeL1(Ω).Notethattheconstanth>0isnotessential,,thesymmetrizationresultcanbesplitintotworesults:(i)The?rstoneappliestorearrangedright-,f2,parisonoftheformf1?f2,thenthesameappliestothesolutions,intheformB(v1)?B(v2).11Forthede?nitionoftheorderrelation?,seeSection7.:..,′(ii)paringthesolutionvof()withanon-rearrangedfunctionfwiththesolutionvcorrespondingtof#,isarearrangedfunctionandB(v#)?B(v),.,B(v)islessconcentratedthanB(v).binedtoobtainsimilarresultsalongthewholechainofiterationsu(tk)oftheevolutionprocess,(parisonandcomparisonofLpnorms)fortheevolutionproblem().Thisapproachcanbeusedinmanydi?,,usuallyoffractionaltype,?rstappliedtoPDEsinvolvingfractionalLaplacianoperatorsin[25],aseisstudied:σ2(?Δ)v=f.()ThispaperusesaninterestingtechniqueofSteinersymmetrizationoftheextendedproblem,basedontheCa?arelli-Silvestreextensionforthede?nitionofσ-[64]bineitwiththeparabolicideasof[58]?c,theydealtwithequationsoftheformσ2?tu+(?Δ)A(u)=f,0<σ<2.()FollowingtheknowntheoryforthestandardLaplacian,thenonlinearityAisanincreasingrealfunctionsuchthatA(0)=0,eptsomeextraregularityconditionsasneeded,likeAsmoothwithA(u)>0forallu>(u)=umwithm>0;theequationisthencalledthefractionalheatequation(FHEforshort)whenm=1,thefractionalporousmediumequation(FPMEforshort)ifm>1,andthefractionalfastdi?usionequation(FFDEforshort)ifm0forallu>,[64–65].WefocusonthelinearcaseA(u)=(?Δ)v+B(v)=f()posedagaininthewholespaceΩ?RNwithzeroDirichletboundaryconditions;h>0isanon-essentialconstant,andthenonlinearityBistheinversefunctiontothemonotonefunctionAthatappearsintheparabolicequation().TheellipticresultsaredevelopedinSections3–paredwiththeresultsof[25,64](v)disappears,andweseth=(theFaber-Krahninequality)Asanapplication,,dueseparatelyto[31,38],basedonaconjecturebyRayleighin1877,,andletBbetheballcenteredattheoriginwithVol(Ω)=Vol(B).Letλ1(Ω)bethe?rsteigenvalueoftheLaplacianoperator,(Ω)≥λ1(B),withequalityifandonlyifΩ=[20](seearecentproofin[15]).Thequestionwewanttoaddresshereisthefollowing:WilltheresultalsoholdfortheusualversionsofthefractionalLap