LECTURE_NOTES_ON_SOBOLEV_SPACES_FOR_CCA

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1、LECTURENOTESONSOBOLEVSPACESFORCCAWILLIEWAI-YEUNGWONG0.1.References.Beforewestart,somereferences:•D.GilbargandN.S.Trudinger,Ellipticpartialdifferentialequationsofsecondorder,Springer.Ch.7.•L.Evans,Partialdifferentialequations,AmericanMath.Soc.Ch.5.•M.E.Taylor,Partialdifferentiale

2、quationsI,Springer.Ch.4.(Note:thispresentationisbasedonheavydosesofFourieranalysisandfunctionalanalysis.)•H.Triebel,Theoryoffunctionspaces,Birkhauser.Ch.2.•R.Adams,SobolevSpaces,AcademicPress.0.2.Notations.WewillworkinRd.p0:givenp≥1arealnumber,wedefinep0tobethepositiverealnumb

3、ersatisfyingp−1+(p0)−1=1;p0iscalledtheHölderconjugateofpΩ:opensetinRd∂Ω:theboundaryofΩ,Ω¯Ω∂α:partialderivativeofmulti-indexα.α=(α1,...,αd)∈(N0)d,withnormPαα1αd

4、α

5、=αi.∂=∂1·∂dΩ1bΩ2:thereexistsacompactsetKsuchthatΩ1⊂K⊂Ω2Dα:weakderivative(see§1.3)ofmulti-indexαsuppf:forafunction

6、f,thisdenotesthesupportset,i.e.thesetonwhichf,0C(Ω):continuousfunctionstakingvalueintherealsdefinedonΩ(thoughmostofwhatwesaywillbevalidforfunctionstakingvalueinaHilbertspace)C(Ω¯):thesubsetofC(Ω)consistingoffunctionsthatextendcontinuouslyto∂ΩC0(Ω):thesubsetofC(Ω¯)consistingoff

7、unctionswhichvanishon∂ΩCk(Ω):functionsfsuchthat∂αf∈C(Ω)forevery

8、α

9、≤k.kisallowedtobe∞(inwhichcasefissmooth)orω(inwhichcasefisanalytic).AnalogouslywedefineCk(Ω¯)andCk(Ω)(notethatthesetCω(Ω)containsonlythezero00functions)Cck(Ω):subsetofCk(Ω)suchthatsuppfbΩppL(Ω),L(Ω):Lebesguespac

10、es(see§1.1)lock,pk,ps,pW(Ω),W(Ω),W(Ω):Sobolevspaces(see§1.4)loc0k·kp:Lpnorm(see§1.1)k·kp,k:Wk,pnorm(see§1.4)VersionasofOctober27,2010.12W.W.WONG1.BasicDefinitionsInthisfirstpartΩcanbetakentobeanyopensubsetofRd.ThroughoutdxwillbethestandardLebesguemeasure.Byameasurablefunctionw

11、e’llmeanarepresentativeofanequivalenceclassofmeasurablefunctionswhichdifferonΩinasetofmeasure0.Thussupandinfshouldbementallyreplacedbyesssupandessinfwhenappropriate.1.1.Lebesguespaces.For∞>p≥1,Lp(Ω)denotesthesetofp-integrablemea-surablefunctions,withnorm1/pZp(1)kukp;

12、Ω=

13、u

14、dx.ΩIfutakesvaluesinsomenormedlinearspace,then

15、·

16、willbe