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1、MAT201CLectureNotes:IntroductiontoSobolevSpacesSteveShkollerDepartmentofMathematicsUniversityofCaliforniaatDavisDavis,CA95616USAemail:shkoller@math.ucdavis.eduMay26,2011Thesenotes,intendedforthethirdquarterofthegraduateAnalysissequenceatUCDavis,shouldbeviewedasaveryshortint
2、roductiontoSobolevspacetheory,andtheratherlargecollectionoftopicswhicharefoundationalforitsdevelopment.ThisincludesthetheoryofLpspaces,theFourierseriesandtheFouriertransform,thenotionofweakderivativesanddistributions,andafairamountofdifferentialanalysis(thetheoryofdif-ferent
3、ialoperators).Sobolevspacesandotherverycloselyrelatedfunctionalframeworkshaveprovedtobeindispensabletopologiesforansweringverybasicquestionsinthefieldsofpartialdifferentialequations,mathematicalphysics,differentialgeometry,harmonicanal-ysis,scientificcomputation,andahostofother
4、mathematicalspecialities.Thesenotesprovideonlyabriefintroductiontothematerial,essentiallyjustenoughtogetgoingwiththebasicsofSobolevspaces.Asthecourseprogresses,Iwilladdsomeadditionaltopicsand/ordetailstothesenotes.Inthemeantime,agoodreferenceisAnalysisbyLiebandLoss,andofcou
5、rseAppliedAnalysisbyHunterandNachtergaele,particularlyChapter12,whichservesasanicecompendiumofthematerialtobepresented.IfonlyIhadthetheorems!ThenIshouldfindtheproofseasilyenough.–BernhardRiemann(1826-1866)Factsaremany,butthetruthisone.–RabindranathTagore(1861-1941)1ShkollerC
6、ONTENTSContents1Lpspaces41.1Notation......................................41.2Definitionsandbasicproperties.........................41.3Basicinequalities.................................5p1.4Thespace(L(X),k·kLp(X)iscomplete....................71.5ConvergencecriteriaforLpfunc
7、tions......................81.6ThespaceL∞(X)................................101.7ApproximationofLp(X)bysimplefunctions.................111.8ApproximationofLp(Ω)bycontinuousfunctions...............111.9ApproximationofLp(Ω)bysmoothfunctions.................121.10Continuouslin
8、earfunctionalsonLp(X)....................141.11AtheoremofF.Riesz..................
