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1、1Appendix:Sobolevspacesandthetracetheorem.1.1SobolevspacesinthewholedomainsSupposethat1·p·1,pisreal.Let•beanon-emptyopensubsetofRn.TheSobolevspaceWpr(•)oforderr2NbasedonthespaceLp(•)isde¯nedbyWr(•):=fu2L(•)=@®u2L(•);forj®j·rgpppwhere®:=(®1;®2;:::;®n),®j2N,isamulti-index
2、,j®j=®1+®2+:::+®nand@®u=@x®@x®:::@x®u.Here@®uisviewedasadistributionon•.Sothecondition@®u2Lp(•)nn¡11meansthatthereexistsafunctionsg®2Lp(•)suchthat=(¡1)j®jforallÁ2D(•):=fu2C1(•);suppu½•g®whereC1(•):=r>0Cr(•).Suchfunctionsg®iscalledtheweakpartialderivativeofu
3、.ThespaceWpr(•)inducedwiththenorm011ZpX@®pAkukWr(•):=j@ujdxp•®·risaBanachspace.Ifp=1,thenwetakeX®kukWr(•):=supj@u(x)j:1x2•®·rRemarkthatweassumedr2N.Tode¯neWpr(•)forr2R,wedenotethesemi-norm:µZZp¶1µZZµ¶p¶1ju(x)¡u(y)jpju(x)¡u(y)jpjuj¹;p;•:=jx¡yjn+p¹dxdy=jx¡yj¹+n=pdxdy;for0
4、<¹<1••••andju(x)¡u(y)jjuj¹;1;•:=sup¹x;y2•jx¡yjwhichiscalledtheHÄoldersemi-norm.Fors=r+¹,0<¹<1,wede¯ne©ªWs(•):=u2Wr(•)=j@®uj<1;forj®j=rpp¹;p;•Ws(•)equippedwiththenormp011pXkuks:=@kukp+j@®ujAWp(•)Wpr(•)¹;p;•j®j=risalsoaBanachspace.Forthecasep=2,weusethenotationHs(•)instea
5、dofWps(•),8s>0;s2R.1A¯rstimportantpropertyofthesespacesisthatforeverys;t2R,s>t>0,theinjectionHs(•)½Ht(•)iscompact.Foreveryreals>0,weset:sk¢kHs(•)H0(•):=D(•)Foreverys>0,wede¯netheSobolevspacewithanegativeorder¡sasthedualspaceofHs(•),i.e.0H¡s(•):=(Hs(•))¤:0ThespaceH¡s(•)i
6、sequippedwithnormofdualspaces:¡sjj8u2H(•);kukH¡s(•):=sup=supjjv2Hs(•)kvkHs(•)s0v2H0(•);kvkHs(•)=1Here,thefunctional:H0s(•)!Cv7!isthedualitymapping,i.e.ude¯nesalinearfunctionalonH0s(•).Asaresult,wehavealsoforeverys;t2R,s>t>0,theinjectionH¡t(•)½H¡s(•)i
7、scompact.1.2Sobolevspacesontheboundaries.Westartbyassumingthat•hasaboundary@•givenbyagraphofclassCk¡1;1,i.e.©ª@•:=(x0;»(x0));x02Rn¡1where»:Rn¡1!RisofclassCk¡1;1(»2Ck¡1and@(k¡1)»hasaboundedderivative).Thinkof•as,forinstance,aperturbedhalfspaceinRn.Ifk=1,•iscalledLipschit
8、z,i.e.ofclassC0;1.Let¡:=@•.Foru2L2(¡),wede¯neu»(x0):=u(x0;»(x0)),forx02Rn¡1.Weset©ªHs(¡):=u2L(¡);u2Hs(Rn¡1);fo
