Sobolev_Spaces,_R.A._Adams,_AP,_1975.pdf
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Sobolev_Spaces,_R.A._Adams,_AP,_1975.pdf
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SOBOLEVSPACESROBERTA.ADAMSDepartmentofMathematicsTheUniversityofBritishColumbiaVancouver,BritishColumbia,CanadaACADEMICPRESSNewYorkSanFranciscoLondon1975ASubsidiaryofHarcourtBraceJovanovich,PublishersCOPYRIGHT(B1975,BYACADEMICPRESS,INC.ALLRIQHTSRESERVED.NOPARTOFTHISPUBLICATIONMAYBEREPRODUCEDORTRANSMITTEDINANYFORMORBYANYMEANS,ELECTRONICORMECHANICAL,INCLUDINGPHOTOCOPY,RECORDINQ,ORANYINFORMATIONSTORAGEANDRETRIEVALSYSTEM,WITHOUTPERMISSIONINWRITINGFROMTHEPUBLISHER.ACADEMICPRESS,INC.111FifthAvenue,NewYak.NewYork10003UnitedKingdomEditionpublishedbyACADEMICPRESS,INC.(LONDON)LTD.24/28OvalRoad,LondonNWlLibraryofCongressCataloginginPublicationDataAdams,RobertASobolevspaces.(Pureandappliedmathematicsseries;v.65)Bibliography:
p.Includesindex.1.Sobolevspaces.I.Title.11.Series:
Pureandappliedmathematics;aseriesofmonographsandtextbooks;v.65QA3.P8QA32351W.8515.774-17978ISBN0-12-044150-0AMS(MOS)1970SubjectClassifications:
46E30,46E35PRINTEDINTHEUNITEDSTATESOFAMERICAPrefaceThismonographisdevotedtoastudyofpropertiesofcertainBanachspacesofweaklydifferentiablefunctionsofseveralrealvariableswhichariseinconnectionwithnumerousproblemsinthetheoryofpartialdifferentialequationsandrelatedareasofmathematicalanalysis,aridwhichhavebecomeanessentialtoolinthosedisciplines.Thesespacesarenowmostoftenasso-ciatedwiththenameoftheSovietmathematicianS.L.Sobolev,thoughtheiroriginspredatehismajorcontributionstotheirdevelopmentinthelate1930s.Sobolevspacesareveryinterestingmathematicalstructuresintheirownright,buttheirprincipalsignificanceliesinthecentralrolethey,andtheirnumerousgeneralizations,nowplayinpartialdifferentialequations.Accordingly,mostofthisbookconcentratesonthoseaspectsofthetheoryofSobolevspacesthathaveprovenmostusefulinapplications.Althoughnospecificapplicationstoproblemsinpartialdifferentialequationsarediscussed(thesearetobefoundinalmostanymoderntextbookonpartialdifferentialequations),thismonographisneverthelessintendedmainlytoserveasatextbookandreferenceonSobolevspacesforgraduatestudentsandresearchersindifferentialequations.SomeofthematerialinChapters111-VIhasgrownoutoflecturenotes181foragraduatecourseandseminargivenbyProfessorColinClarkattheUniversityofBritishColumbiain1967-1968.Thematerialisorganizedintoeightchapters.ChapterIisapotpourriofstandardtopicsfromrealandfunctionalanalysis,included,mainlywithoutXiXiiPREFACEproofs,becausetheyformanecessarybackgroundforwhatfollows.ChapterI1isalsolargely“background”butconcentratesonaspecifictopic,theLebesguespacesLP(R),ofwhichSobolevspacesarespecialsubspaces.Forcompleteness,proofsareincludedhere.Mostofthematerialinthesefirsttwochapterswillbequitefamiliartothereaderandmaybeomitted,orsimplygivenasuperficialreadingtosettlequestionsofnotationandsuch.(PossibleexceptionsareSections1.25-1.27,1.31,and2.21-2.22whichmaybelessfamiliar.)Theinclusionoftheseelementarychaptersmakesthebookfairlyself-contained.Onlyasolidundergraduatebackgroundinmathematicalanalysisisassumedofthereader.ChaptersIII-VImaybedescribedastheheartofthebook.ThesedevelopallthebasicpropertiesofSobolevspacesofpositiveintegralorderandculminateintheveryimportantSobolevimbeddingtheorem(Theorem5.4)andthecorrespondingcompactimbeddingtheorem(Theorem6.2).Sections5.33-5.54and6.12-6-50consistofrefinementsandgeneralizationsofthesebasicimbeddingtheorems,andcouldbeomittedfromafirstreading.ChapterVIIisconcernedwithgeneralizationofordinarySobolevspacestoallowfractionalordersofdifferentiation.Suchspacesareofteninvolvedinresearchintononlinearpartialdifferentialequations,forinstancetheNavier-Stokesequationsoffluidmechanics.Severalapproachestodefiningfractional-orderspacescanbetaken.WeconcentrateinChapterVIIonthetrace-interpolationapproachofJ.L.LionsandE.Magenes,anddiscussotherapproachesmorebrieflyattheendofthechapter(Sections7.59-7.74).Itisnecessarytodevelopareasonablebodyofabstractfunctionalanalysis(thetrace-interpolationtheory)beforeintroducingthefractional-orderspaces.Mostreaderswillfindthatareadingofthismaterial(inSections7.2-7.34,possiblyomittingproofs)isessentialforanunderstandingofthediscussionoffractional-orderspacesthatbeginsinSection7.35.ChapterVIIIconcernsOrlicz-Sobolevspacesand,forthesakeofcom-pleteness,necessarilybeginswithaself-containedintroductiontothetheoryofOrliczspaces.Thesespacesarefindingincreasinglyimportantapplicationsinappliedanalysis.ThemainresultsofChapterVIIIarethetheoremofN.S.Trudinger(Theorem8.25)establishingalimitingcaseoftheSobolevimbed-dingtheorem,andtheimbeddingtheoremsofTrudingerandT.K.DonaldsonforOrlicz-SobolevspacesgiveninSections8.29-8.40.TheexistingmathematicalliteratureonSobolevspacesandtheirgeneral-izationsisvast,anditwouldbeneithereasynorparticularlydesirabletoincludeeverythingthatwasknownaboutsuchspacesbetweenthecoversofonebook.Anattempthasbeenmadeinthismonographtopresentallthecorematerialinsufficientgeneralitytocovermostapplications,togivethereaderanoverviewofthesubjectthatisdifficulttoobtainbyreadingresearchpapers,andfinally,asmentionedabove,toprovideareadyreferenceforPREFACExiiisomeonerequiringaresultaboutSobolevspacesforuseinsomeapplication.Completeproofsaregivenformosttheorems,butsomeassertionsareleftfortheinterestedreadertoverifyasexercises.Literaturereferencesaregiveninsquarebrackets,equationnumbersinparentheses,andsectionsarenumberedintheformm.nwithmdenotingthechapter.AcknowledgmentsWeacknowledgewithdeepgratitudetheconsiderableassistancewehavereceivedfromProfessorJohnFournierinthepreparationofthismonograph.AlsomuchappreciatedarethehelpfulcommentsreceivedfromProfessorBuiAnTonandtheencouragementofProfessorColinClarkwhooriginallysuggestedthatthisbookbewritten.ThanksarealsoduetoMrs.Yit-SinChooforasuperbjoboftypingadifficultmanuscript.Finally,ofcourse,weacceptallresponsibilityforerrororobscurityandwelcomecomments,orcorrections,fromreaders.xvListofSpacesandNormsThenumbersattherightindicatethesectionsinwhichthesymbolsareintroduced.Insomecasesthenotationsarenotthoseusedinotherareasofanalysis.II*Ile7.2I1*;B1+B2II7.11II*;BSPP(R)II7.67II*;Bs*p(a)ll7.721.251.251.251.251.251.261.275.28.371.511.527.7,7.91.11.1xviiiLlSTOFSPACESANDNORMS7.298.143.13.127.338.71-40,1.461.532.13.12.55.227.37.37.57.627.668.91.17.147.24,7.26,7.327.327.357.433.13.13.116.547.117.307.36,7.397.397.487.498.278.278.271.4,1.6,6.511.5,1.101.22IIntroductoryTopicsNotation1.1ThroughoutthismonographthetermdomainandthesymbolRshallbereservedforanopensetinn-dimensional,realEuclideanspaceR.WeshallbeconcernedwithdifferentiabilityandintegrabilityoffunctionsdefinedonQ-thesefunctionsareallowedtobecomplexvaluedunlessthecontraryisstatedexplicitly.Thecomplexfieldisdenotedby.ForcEandtwofunctionsuanduthescalarmultiplecu,thesumu+11,andtheproductuuarealwaystakentobedefinedpointwiseas(4(4=cu(x),(u+u)(x)=u(x)+u(x),(uu)(x)=u(x),atallpointsxwheretherightsidesmakesense.AtypicalpointinRisdenotedbyx=(xl,.,x,);itsnorm1x1=(z=,x).Theinnerproductofxandyisx.y=If01=(a1,.,E,)isann-tupleofnonnegativeintegersaj,wecallaamulti-indexanddenotebyxuthemonomialx;.ex:
whichhasdegreela1=Cj=aj.Similarly,ifDj=i?
/axjfor1sj-,f,9EX,xEX,cEQ=.XwillbeaTVSprovidedasuitabletopologyisspecifiedforit.Onesuchtopologyistheweak-startopology,theweakesttopologywithrespecttowhichthefunctionalF,definedonXbyFcf)=f(x)foreachfEXiscontinuousforeachxEX.Thistopologyisused,forinstance,inthespaceofSchwartzdistributionsintroducedinSection1.52.Thedualofanormedspacecanbegivenastrongertopologywithrespecttowhichitisanormedspaceitself(Section1.10).NormedSpaces1.6AnormonavectorspaceXisareal-valuedfunctionalfonXwhichsatisfies(i)f(x)20forallxEXwithequalityifandonlyifx=0,(ii)f(cx)=Iclf(x)foreveryxEXandcE,(iii)f(x+y)sf(x)+f(y)foreveryx,yEX.41INTRODUCTORYTOPICSAnormedspaceisavectorspaceXwhichisprovidedwithanorm.Thenormwillbedenotedby11.;X11exceptwheresimplernotationsareintroduced.Ifr0,thesetBJx)=yEX:
Ily-x;X(I0suchthatB,(x)cA.TheopensetsthusdefinedconstituteatopologyforXwithrespecttowhichXisaTVS.ThistopologyiscalledthenormtopologyonX.TheclosureofE,(x)inthistopologyisB,(x)=yEX:
(Iy-x;XII5r.ATVSXisnormableifitstopologycoincideswiththetopologyinducedbysomenormonX.TwodifferentnormsonavectorspaceareequivalentiftheyinducethesametopologyonX,thatis,ifforsomeconstantc0-cIIxII15llxllz5(/c)llxll1forallxEX,)IIfXandYaretwonormedspacesandifthereexistsaone-to-onelinearoperatorLmappingXontoYandhavingtheproperty11L(x);Y11=Ilx;X11foreveryxEX,thenLiscalledanisometricisomorphisnibetweenXandY,andXandYaresaidtobeisometricallyisomorphic;wewriteXEY.Suchspacesareoftenidentifiedsincetheyhaveidenticalstructuresanddifferonlyinthenatureoftheirelements.and11.I2beingthetwonorms.1.7Asequencex,inanormedspaceXisconvergenttothelimi
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