Evans-PDE-Solution-Chapter-5-Sobolev

PartialDierentialEquations,2ndEdition,L.C.EvansChapter5SobolevSpacesShih-HsinChen,Yung-HsiangHuangy2016/11/011.Proof.Weprovethecasek=0forsimplicity.Thegeneralcaseisalmostthesame(exceptweneedthestandardconvergencetheorembetweenffngandfDfng).GivenaCauchysequenceffngC0;(U),thenthereisaf2C(U)suchthatfn!funiformlyandaconstantM0suchthatforeachn2Nands6=t,jfn(s)fn(t)jjstjM:Thenforeachs6=t,thereissomeN=N(s;t)2Nsuchthatjf(s)f(t)jjf(s)fN(s)j+jfN(s)fN(t)j+jfN(t)f(t)j2jstj+Mjstj:Therefore,f2C0;(U).Itremainstoshowfn!finC0;(U).Foreachs;t2Uand0,byCauchy'scriteria,wecanndaN=N()suchthatfork;nNjfk(s)fn(s)fk(t)+fn(t)jjstjForeach0,wecanndaK=K(;s;t)N,suchthatjf(x)fK(x)jforx=s,orx=t.Therefore,foreverynNjf(s)fn(s)f(t)+fn(t)jjf(s)fK(s)j+jfK(s)fn(s)fK(t)+fn(t)j+jf(t)fK(t)j2+jstj:Letting!0,weobtainthatjf(s)fn(s)f(t)+fn(t)jjstjforallnN().DepartmentofMath.,NationalTaiwanUniversity.Email:[e-mail].,NationalTaiwanUniversity.Email:[e-mail]0;=supju(x)j+supju(x)u(y)jjxyj.Setts.t.(1t)+t=.Thenju(x)u(y)jjxyj=ju(x)u(y)jjxyj1tju(x)u(y)jjxyjt:kukC0;kuk1t1kukt1+[u]1tC0;[u]tC0;1:=a1t1bt1+a1t2bt2:Sincey=xt,0t1,isaconcavefunction,wehavea1t1bt1+a1t2bt2=(a1+a2)a1a1+a2b1a1t+a2a1+a2b2a2t(a1+a2)b1a1+a2+b2a1+a2t=(a1+a2)1t(b1+b2):ThenkukC0;kuk1t1kukt1+[u]1tC0;[u]tC0;1(kuk1+[u]C0;)1t(kuk1+[u]C0;1)t=kuk1tC0;kuktC0;1:Remark0.1.ThereisageneralconvexitytheoremforHigher-orderHoldernorm(duetoHormander),seeHelms[3,chapter8].3.Omit.4.Proof.(b)followseasilyfrom(a)andHolderinequality.For(a),denotetheweakderivativeofubyu0w.Denev(t)=Zt0u0w(s)ds,thenvisabsolutelycontinuous.Given2C1c,Z10[vu]0dx=Z10Zx0u0w(t)dt0(x)dxZ10u(x)0(x)dx=Z10Z1t0(x)dxu0w(t)dt+Z10u0w(x)(x)dx=Z10(t)u0w(t)dt+Z10u0w(x)(x)dx=0:(ThisargumentavoidstheuseoftheLebesgueDierentiationtheorem.)WearedoneifweapplythefollowingtheoremfromDistributiontheorywithapartialproof,thewholeproofcanbefoundinLiebandLoss[5,Section6.9-6.11].Theorem0.2.LetTbeadistributiononanopenconnectedsetinRn.IfalltherstdistributionalderivativesofTare0,thenTisaconstantfunction.2Proof.(TakenfromWheedenandZygmund[8,Page464],weassumeT2L1loc().)Letk(x)bethestandardmollier.GivenBbeasmallballcontainedin.Forsmall,byhypothesis,weseethatforeachy2B,0=ZT(x)@@xi[k(yx)]dx=@@yiZT(x)[k(yx)]dx;byLDCTHence,forallsmall,thereisaconstantcsuchthat(weomitsomeirrelevantdetails)ZT(x)k(yx)dx=cforally2B:As!0,thisintegralconvergestoTinL1(B),henceitconvergestoT(y)fora.e.y(uptosubsequence),andconsequentlycconvergestosomeconstantc,andT(y)=ca.e.inB.Sinceispathconnected,Tisconstanta.e.in.Remark0.3.IthinkthereadershouldtrytoprovethegeneralcasethatTisadistributionbymodifyingtheaboveproof.5.Proof.ForanyVWU,wecanndasmoothfunctionusuchthatu1onVandsupp(u)W.Take=13dist(V;Wc)0.Let(z)bethestandardmolliersupportedinfjzjg;V0:=fz:dist(z;V)gandu(x):=V0(x)=ZRnV0(xy)(y)dy=ZxV0(y)dy:Thenuissmooth(byLDCT),u=1onVandsupportedinfdist(x;V)2gW.6.ApplyExercise5.WeomititsinceitcanbefoundinmanyDierentialGeometrytextbooks.7.Proof.Z@UjujpdSZ@Ujujp
dS=ZUdiv(jujp)dx=ZUjujpdiv+r(jujp)
dx38.Proof.Considerun(x)=h1ndist(x;@U)i+.Notethatun2C(U);0un(x)1andfun(x)gisdecreasingto0foreachx2U,MCTtellsuskunkp;U!0.Ontheotherhand,kunkp;@U=k1kp;@Uisapositiveconstant.HencekTunkp;Ukunkp;U=kunkp;@Ukunkp;U!1whichmeansTwillnotbebounded,ifTexists.9.Proof.Givenu2H2(U)\H10(U),thereexistsuk2C1(U)withuk!uinH2(U),andvk2C1c(U)withvk!uinH1(U).Hencefrukgandf
ukgareboundedinL2.NoteZ@Uuk@uk@ndS=ZUdiv(ukruk)dx=ZUjrukj2dx+ZUuk
ukdx:Theright-handsidetendstoZUjruj2dx+ZUu
udx.NoteZ@Uuk@uk@ndS=ZUdiv((ukvk)ruk)dx=ZUr(ukvk)ruk+(ukvk)
ukdx:whichtendsto0byHolderand(ukvk)!0inH1(U).Therefore,forallu2H2(U)\H10(U),ZUjruj2+u
udx=0ThenweapplyHolder'sinequalitytogetthedesiredinequality.10.Proof.(Sketch)(b)issimilarto(a).Toprove(a),westartwiththeformulainHint,dointegrationbypartsonce,andthenapplythegeneralizedHolderwithexponentspair(p;p;pp2).11.Proof.GivenKU,compact.Thenu2L1(K).NotethatD(u)=Du=0.SinceUisconnected,u=cwhichconvergestouinLpandhencetou(y)forsomey(uptosubsequence),andconsequentlycconvergestosomeconstantc.Therefore,u=constanta.e.inKandhenceinU.12.Proof.ConsiderinR1,letu(x)=(0;1)(x).13.Proof.ConsiderU=B(0;1)f(x;y)x0ginR2,letu(x;y)=rinpolarcoordinate.14.Proof.ZUju(x)jndx=!nZ10loglog1+1rnrn1dr;lett=1r=!nZ111tn
jloglog(1+t)jntdt4Sincelimt!1jloglog(1+t)jnt=0,thereisT0suchthatjloglog(1+t)jnt1fortT.ThenZUju(x)jndx=!nZ111tn
jloglog(1+t)jntdt!nZT11tn
jloglog(1+t)jntdt+!nZ111tndt1:Inaddition,uxi=1log(1+1jxj)
11+1jxj
xijxj3:ThenZUjuxijndx!nZ101log(1+1r)
11+1r
1r2nrn1dr;lett=1r=!nZ111jlog(1+t)jn
1(1+t)n
tn1dt!nnZ1log21sn
1esn
esnds(s=log(1+t))1:15.Proof.ApplyingPoincareinequality,ZUu2dx12ZU(u)2Udx12+ZUju(u)Uj2dx12(u)UjUj12+CZUjDuj2dx12:ApplyingHoliderinequality,(u)U=1jUjZfu6=0gudx1jUjZfu6=0g12dx12Zfu6=0gu2dx12=1jUj(jUj)12ZUu2dx12:Therefore,ZUu2dx12