topology & sobolev spaces - brezis

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1、TOPOLOGYANDSOBOLEVSPACESHaimBrezis,(1);(2),andYanYanLi,(2)Section0.IntroductionLetMandNbecompact1connectedorientedsmoothRiemannianmanifoldswithorwithoutboundary.ThroughoutthepaperweassumethatdimM2butdimNcouldpossiblybeone,forexampleN=S1isofinterest.Ourfunctionalframewor

2、kistheSobolevspaceW1;p(M;N)whichisdenedbyconsideringNassmoothlyembeddedinsomeEuclideanspaceRKandthenW1;p(M;N)=fu2W1;p(M;RK);u(x)2Na:e:g;1;pwith1p<1.W(M;N)isequippedwiththestandardmetricd(u;v)=ku−vkW1;p.OurmainconcernistodeterminewhetherornotW1;p(M;N)ispath-connectedand

3、ifnotwhatcanbesaidaboutitspath-connectedcomponents,i.e.itsW1;p-homotopyclasses.WesaythatuandvareW1;p-homotopicifthereisapathut2C([0;1];W1;p(M;N))suchthatu0=uandu1=v.Wedenotebythecorrespondingequivalencerelation.LetdenoteptheequivalencerelationonC0(M;N),i.e.uvifthereis

4、apathut2C([0;1];C0(M;N))suchthatu0=uandu1=v.FirstaneasyresultTheorem0.1.AssumepdimM,thenW1;p(M;N)ispath-connectedifandonlyifC0(M;N)ispath-connected.Theorem0.1isbasicallyknown(andreliesonanideaintroducedbySchoenandUh-lenbeck[SU]whenp=dimM;seealsoBrezisandNirenberg[BN]).O

5、necanalsodeduceitfromPropositionsA.1,A.2andA.3intheAppendix.Since,ingeneral,C0(M;N)isnotpath-connected,thismeansthatW1;p(M;N)isnotpath-connectedwhenpisarge".Ontheotherhandifpismall",weexpectW1;p(M;N)tobepath-connectedforallMandN.Indeedwehave1SeeRemarkA.1intheAppendixifNi

6、snotcompact.12SECTION0.INTRODUCTIONTheorem0.2.Let1p<2(andrecallthatdimM2).ThenW1;p(M;N)ispath-connected.OurproofofTheorem0.2issurprisinglyinvolvedandrequiresanumberoftechnicaltoolswhicharepresentedinSections1-4.Wecalltheattentionofthereaderespeciallytothebridging"meth

7、od(seeProposition1.2andProposition3.1)whichisnewtothebestofourknowledge.Remark0.1.Assumption1p<2inTheorem0.2issharp(forgeneralMandN).Forexampleifisanyopenconnectedset(oraconnectedRiemannianmanifold)ofdimension1,thenW1;2(S1;S1)isnotpath-connected.Thismaybeseenusingth

8、eresultsofB.White[W2]orRubinstein-Sternberg[RS].Thisisalsoaconsequenceoftheresultin[BLMN]whichwerecallf