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isprovedWenowcansettletheposedquestioncompletelyasacorollaryofourmaint
isprovedWenowcansettletheposedquestioncompletelyasacorollaryofourmaint

isprovedWenowcansettletheposedquestioncompletelyasacorollaryofourmaint - Description


D2RmInTheorem11somR2denotesthespaceofskewsymmetricm2mmatriceswithentriesinR2rx1x2tisthegradientand1rustandsforthematrixproductwithentriesgivenbythescalarproductoftherespectivecomponentsofandruRemark ID: 871182 Download Pdf

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1 isproved.Wenowcansettletheposedquestionc
isproved.Wenowcansettletheposedquestioncompletelyasacorollaryofourmaintheorem:Theorem1.1Let 2L2(D2;som R2)ande2Ls(D2;Rm),s�1,begiven.Then,anyweaksolutionu2W1;2(D2;Rm)of�4u= ru+einD2(1.2)belongstoC0; (D2;Rm)forsome �0.Ifthetraceu @D2iscontinuous,thenweconcludeu2C0; (D2;Rm)\C0( D2;Rm).InTheorem1.1,som R2denotesthespaceofskew-symmetricmm-matriceswithentriesinR2,r=(@x1;@x2)tisthegradientand rustandsforthematrixproductwithentriesgivenbythescalarproductoftherespectivecomponentsof andru.Remark1.2ThemainnewcontributioninTheorem1.1isthecontinuityre-sultuptotheDirichlettypeboundary.TheinteriorregularitywasprovedbyT.Rivierein[Riv07](fore0)andourproofisbasedonRiviere'sdecomposi-tionresultcombine

2 dwiththeDirichletgrowthapproachbyRivi
dwiththeDirichletgrowthapproachbyRiviereandStruwein[RS08]aswellassomeadditionalargumentsduetoP.Strzelecki[Str03].Remark1.3LetusemphasizethatonecanproveTheorem1.1alsobyre ec-tionacross@D2,wheneverthereissome 2W2;p(D2;Rm),p�1,suchthatu= on@D2.Indeed,thedi erencefunctionv:=u� 2W1;20(D2;Rm)thenalsosolvessystem(1.2)withazeroorderterm~e2L~s(D2;Rm)forsome~s�1.Oddre ectionofvandappropriatere ectionofthedata ;~ethenyieldsananaloguesystemonsomelargerdiscB1+(0),�0,andtheassertionfollowsbyRiviere'sinteriorregularityresult.Remark1.4InTheorem1.1theunitdiscD2canbereplacedbyanyothersimplyconnecteddomainR2withC1; -boundary, �0,accordingtotheRiemannianmappingtheoremandthewellestablishedboundary

3 behaviourofconformalmappings;seeforinsta
behaviourofconformalmappings;seeforinstance[Pom92]Chapter3.ReturningtoHeinz'conjecturementionedabove,weobtainthefollowingCorollary1.5LetH2L1(R3)begivenandletu2W1;2(D2;R3)beaso-lutionof(1.1)withcontinuousboundarytraceuj@D2.Thenthereholdsu2C0; (D2)\C0( D2)forsome �0.ThisfollowsdirectlyfromTheorem1.1bywriting(1.1)intheform(1.2)with :=H(u)0@0r?u3�r?u2�r?u30r?u1r?u2�r?u101A2L2(D2;so3 R2);whereweabbreviatedr?:=(�@x2;@x1)andu=(u1;u2;u3).LetusemphasizethatTheorem1.1canbeapplied,moregenerally,tosta-tionarypointsofconformallyinvariantfunctionalsintwodimensions.HavingGruter's[Gru84]characterizationinmind,wecangivethefollowinggeomet-ricdescription(seee.g.[Cho95]and[Riv07]fordetails):LetNbeasmooth2 andkrPkL2(B2R0(x0))+krkL2(B2

4 R0(x0)).(2.3)Nowconsideraweaks
R0(x0)).(2.3)Nowconsideraweaksolutionu2W1;2(D2;Rm)of�4u= ru+einD2asinTheorem1.1.Formula(2.2)thenyields�div(P�1ru)=r?P�1ru+P�1eweaklyinB2R0(x0)\D2:(2.4)Next,letx12D2and%�0bechosenwithB2%(x1)BR(x0).Then,alinearHodge-decomposition1givesusP�1ru=rf+r?g+hinB%(x1)(2.5)withfunctionsf;g2W1;20(B%(x1);Rm)andaharmonich2L2(B%(x1);Rm R2)\C1(B%(x1);Rm R2).Inaddition,wehave�4f=�div(P�1ru)=r?P�1ru+P�1einB%(x1),�4g=�curl(P�1ru)=r?P�1ruinB%(x1).(2.6)Forr2(0;%)andp2(1;2)wenowcanestimateZBr(x1)jrujpdx=ZBr(x1) P�1ru pdx(2.5)CpZBr(x1)�jhjp+jrfjp+jrgjp:(2.7)Consequently,MorreytypeLp-estimatesforf;g;hwillyieldsuchestimatesforthesolutionu.2.2Morreyt

5 ypeestimatesPickx12D2and%�0withB2
ypeestimatesPickx12D2and%�0withB2%(x1)D2.De nev%:=(u�(u)x1;%)(2.8)fortheu2W1;2(D2;Rm)from(2.6),where2C10(B3 2%(x1))issomecut-o functionsatisfying01,1onB%(x1)andjrjC %.Thenwehavethefollowingcrucialestimates:Lemma2.2Letf;g2W1;20(B%(x1);Rm)besolutionsof(2.6)forsomegivenP2W1;2(B%(x1);SOm),2W1;2(B%(x1);som)satisfying(2.3)withsmall�0,aswellassomeu2W1;2(D2;Rm)ande2Ls(B%(x1);Rm)withs�1.Then,foreveryp2(1;2),thereisaconstantCCp;ssuchthatkrfkLp(B%(x1))+krgkLp(B%(x1))C%2 p�1[v%]BMO+C%1+2 p�2 skekLs(B%(x1)):(2.9)Furthermore,foranyharmonich2Lp(B%(x1))andanyr2(0;%],thereholdsZBr(x1)jhjpCpr %2ZB%(x1)jhjp: 1Forarbitrary2L2(D2;Rm R2),

6 de nef;g2W1;20(B%(x1);Rm)asweaksolut
de nef;g2W1;20(B%(x1);Rm)asweaksolutionsof4f=div()and4g=curl().Thenh:=�rf�r?gisharmonicinD2.5 Puttingeverythingtogetherandusingk'kW1;q1aswellasthede nition(2.8)ofv%,weprovedtheestimateforfin(2.9).Forgwecalculate5ZB%(x1)rgr'(2.6)=ZB%(x1)r?P�1ru'=ZB%(x1)r?P�1r'[(u�(u)x0;%)]kr?P�1r'kH[v%]BMOCkrP�1kL2(B%(x1))kr'kL2(B%(x1))[v%]BMO(2.12)CkrP�1kL2(B%(x1))%2 p�1kr'kLq(B%(x1))[v%]BMOCkrP�1kL2(B%(x1))%2 p�1[v%]BMO:TheconstantintheHardy-estimateisindependentof%,asonecanseebyscaling.Hence,wealsoobtainedtheestimateforgin(2.9).Finally,weestablishtheestimatefortheharmonictermh.EstimatingasinTheorem2.1onp.78ofGiaquinta'smonograph[Gia83]andapplyingtheembeddingW2;p

7 ,!L1aswellasLp-theory,we nd:Foranyh2
,!L1aswellasLp-theory,we nd:Foranyh2Lp(B1(0)),p�1,with4h=0inB1(0)andforany 2(0;1 2)holdsZB (0)jhjpCp 2khkpL1(B1 2(0))Cp 2khkpW2;p(B1 2(0))Cp 2khkpLp(B1(0)):Ofcourse,thisresultremainsvalidalsofor1 1 2.Hence,uponshiftingtheinequalitytosomex1andscaling,weinferZBr(x1)jhjpCpr %2ZB%(x1)jhjpforeveryharmonich2Lp(B%(x1)),p�1,andforevery0r%.Thiscompletestheproof.2.3CompletionoftheproofofTheorem1.1LettheassumptionsofTheorem1.1besatis ed,thatis:Consideraweaksolutionu2W1;2( ;Rm)of�4u= ru+einD2(2.13)withcontinuoustraceuj@D2.Here, 2L2(D2;som R2)ande2Ls(D2;Rm),s&#x-458;1,aregiven,andw.l.o.g.wemayassumes2(1;4 3). 5Herewehavethesameproblemasforthecalculationsoff,whichwesolvebyapproxi-mati

8 onofP,cf.footnote3,page7.8 Wereturntothe
onofP,cf.footnote3,page7.8 WereturntothesituationdescribedinSubsection2.1:Choosinganarbi-trary2(0;"m]andR0=R0()2(0;1)suitablysuchthat(2.1)isful lled,wepickx02D2andR�0withRminf1�jx0j;R0g.Foranyx12D2and%&#x]TJ/;ø 9;&#x.962; Tf;&#x 21.;ɗ ;� Td;&#x [00;0withB2%(x1)BR(x0)wethenfoundfunctionsf;g2W1;20(B%(x1);Rm),whichsolve(2.6)withP;fromLemma2.1,andsomeharmonicfunctionh2L2(B%(x1);Rm R2)suchthattheestimate(2.7)isful lledforanyr2(0;%)andanyp2(1;2).CombiningthisinequalitywithLemma2.2fromSection2.2,wearriveatZBr(x1)jrujpCpr %2ZB%(x1)jhjp+CpZB%(x1)(jrfjp+jrgjp)(2.5)Cpr %2ZB%(x1)jrujp+CpZB%(x1)(jrfjp+jrgjp)(2.9)Cpr %2ZB%(x1)jrujp+Cp%2�p[v%]pBMO+Cp;s%2�

9 ;p+2p�2p skekpLs(B%(x1)):Multiplyingt
;p+2p�2p skekpLs(B%(x1)):Multiplyingthisbyrp�2andde ningJp(a;r;u):=1 r2�pZBr(a)jrujp;Mp(a;r;u):=supz2Br(a);%�ja�zj1 %2�pZB%(z)jrujp;weinferJp(x1;r;u)Cpr %pJp(x1;%;u)+Cpr %p�2[v%]pBMO+Cp;sr %p�2%2p(1�1 s)kekpLs(B%(x1))forall0r%andx12D2withB2%(x1)BR(x0).Inordertoexploitthislastrelation,wehavetoestimate[v%]BMOappropriately.ThiscanbedonebyexactlythesamecalculationsasinStep5ofStrzlecki'sarticle[Str03]:Proposition2.3ThereisaconstantCpsuchthat[v%]BMOCpMp(x1;2%;u)1 pforallx12D2and%�0withB2%(x1)D2.Now,picksome 1tobe xedlaterandsetr:= %.Then,forallx12D2and%&#x]TJ/;ø 9;&#x.962; Tf;&#x 19.;؈ ;� Td;&#x [00;0withB2%(x1)BR(x0),weconcludeJp(x1; %;

10 u)Cp p(1+ �2)Mp(x0;R;u)+Cp;
u)Cp p(1+ �2)Mp(x0;R;u)+Cp;s p�2%2p(1�1 s)kekpLs(BR(x0)):TheconstantCpisindependentofR0andhenceof.Wechoose 1smallenoughtoensureCp p1 4.Setting:=min( 2;"),wegettheestimateJp(x1; %;u)1 2Mp(x0;R;u)+Cs;p; R2p(1�1 s)kekpLs(BR(x0))9 Proposition2.4(Dirichletgrowththeorem)ThereisaconstantCsuchthat,forall%2(0;R0),a2D2withB%(a)D2andforanysolutionu2W1;2(D2)of(2.15),theinequalityju(x)�u(y)jCp (krukL2(B%(a))+kekLs(B%(a)));x;y2B% 2(a)(2.16)holdstrue.Forconvenience,wesketchtheproofofProposition2.4inSubsectionA.3.Now,havingtheestimate(2.16)forthemodulusofcontinuityforoursolutionu2W1;2(D2)of(2.13)inmindandassumingthecontinuityofuj@D2,thedesiredglobalregularityu2C0( D2;Rm)followsfromthefollowinglemmabyStrzeleck

11 i:Lemma2.5(Strzelecki,2003)(c.f.[Str03],
i:Lemma2.5(Strzelecki,2003)(c.f.[Str03],lemma3.1)Letu2W1;2(D2;Rm)\C0(D2;Rm).AssumethatthereareR0�0andamappingF:D2(0;R0)!(0;+1)suchthatwehaveju(x)�u(y)jF(a;%)forallx;y2B% 2(a)(2.17)forany%2(0;R0),a2D2withB%(a)D2.IfF(;%)%!0���!0uniformlyinD2andifthetraceofuon@D2iscontinuous,thenwe ndu2C0( D2;Rm).WerecalltheproofofthislemmainSubsectionA.4.TheproofofTheorem1.1iscompleted.AAppendixFortheconvenienceofthereaderwewill rststatesomeresultsfromharmonicanalysisand,asacorollary,partofWente'sinequality,whichwewilluseaf-terwardstosketchtheproofoftheUhlenbeck-Rivieredecompositionofsomeskew-symmetric .Inaccordancewiththeirapplicationsinthepresentpaper,allresultsarestatedontwo-dimensionaldiscs,exceptforthede

12 ;nitionsandbasicpropertiesofHardy-andBMO
;nitionsandbasicpropertiesofHardy-andBMO-spaces.Nevertheless,someresultsextendintheirspirittohigherdimensions.A.1SomefactsfromHarmonicAnalysisandWente'sIn-equalityWestartwiththede nitionsofBMOandtheHardy-spaceH.Formoredetailsandproofswerefer,e.g.,toStein'smonograph[Ste93].ForapplicationsofHardyspacestoPDEtheorytheinterestedreadermayconsideralsoSemmes'article[Sem94].De nitionA.1(BMOandHardy-space)LetTdenotethesetoftestfunc-tions2C10(B1(0))withjrj1everywhereinB1(0).De netheHardyspaceHasthespaceofallfunctionsf2L1(Rn)havingtheirassociatedmaxi-malfunctionf(x):=sup�t0sup2T ZRn1 tnx�y tf(y)dy 11 inL1(Rn).ThenormiskfkH:=kfkL1(Rn):Thespaceofboundedmeanosc

13 illationBMOisthespaceofallf2L1loc(Rn)suc
illationBMOisthespaceofallf2L1loc(Rn)suchthat[f]BMO=supx2Rnr�0�ZBr(x)jf�(f)x;rj1istruewith(f)x;r=�ZBr(x)f:Motivatedbytheresultsof[Mul90],Coifman,Lions,MeyerandSemmesprovedin[CLMS93]thefollowingTheoremA.2(Hardyspacesanddiv-curl-terms)Let1p;q1with1 p+1 q=1bechosen.LetA2Lp(Rn;Rn)andB2Lq(Rn;Rn)beweaksolutionsofdiv(A)=0andcurl(B)=0inRn:ThenwehaveAB2HandtheestimatekABkHkAkLp(Rn)kBkLq(Rn)istrue.Thefollowingduality-liketheoremwasobtained rstin[FS72]:TheoremA.3(BMO-Hardy-duality)ThereexistsaconstantCndepend-ingonlyonthedimensionn,suchthatforeverysmoothf2BMO(Rn)andg2H(Rn)thefollowinginequalityholds ZRnfg Cn[f]BMOkgkH:TheoremA.4(Wente'sinequality)(c.f.[Wen69],[Tar85],[BC84])Leta2W1;

14 2(D2),b2W1;p(D2)begivenwithsomep2(1;1)an
2(D2),b2W1;p(D2)begivenwithsomep2(1;1)andletu2W1;2(D2)beaweaksolutionof(�4u=rar?binD2;u=0on@D2:(A.1)ThenubelongstoW1;p(D2)andwehavetheinequalitykrukLp(D2)CpkrakL2(D2)krbkLp(D2):Proof.Thetheoremfollowsbycompactness,ifwecanproveitfora2C1( D2);�ZD2a=0;b2C1( D2):Furthermore,weassumeaandbtobeextendedtofunctionswithcompactsupportinW1;2(R2)andW1;p(R2),respectively.Letq=p p�1betheconjugatedexponentofp.WritingX=C10(D2;R2),wecalculatekrukLp(D2)=ZD2rujrujp�2ru krukp�1Lp(D2)supF2XkFkLq(D2)1ZD2ruF:12 BylinearHodgedecomposition,wecansplitanyF2XintoF=r'+h;where'2W1;20(D2)andh2L2(D2)satis esZD2ruh=0:ByLq-Theorywehave6kr'kLq(D2)CqkFkLq(D2):Hence,wearriveatkrukLp(D2)Cqsup'2Y;kr'kLq(D2)1ZD2rur';wherewe

15 abbreviatedY=C10(D2).ApplyingtheBMO-Hard
abbreviatedY=C10(D2).ApplyingtheBMO-Hardy-Duality,Theo-remA.3,to(A.1),andthenusingtheextensionoperator,Holder-andPoincareinequality,weobtainforany'2Y:ZD2rur'=ZD2rar?b'=�ZD2ar?br'=�ZR2(a��ZD2a)r?br'C[a]BMOkr?(b��ZD2b)r('��ZD2')kHCkrakL2(D2)krbkLp(D2)kr'kLq(D2);whichcompletestheproof.Itisclear,thatthistypeofproofdoesextendtohigherdimensionsaswellastothecaseofhomogeneousNeumannboundarydata.A.2Decompositionofrealskew-symmetricMatricesWesketchheretheproofofLemma2.1.ThisresulthasbeenprovedbyRivierein[Riv07],adaptingthetechniquesbyUhlenbeck,whoprovedasimilarresultin[Uhl82].Lemma2.1followsbycompactnessfromthefollowingLemmaA.5Thereareconstants"m�0andCm�0suchthatthefollowingh

16 olds:Let 2W1;2(D2;som R2)begivenwithk kL
olds:Let 2W1;2(D2;som R2)begivenwithk kL2(D2)"m:(A.2) 6HereweusekgkW1;p0Ck4gk(W1;p0)whichistrueforp2(seeforexample[GM05],Theorem7.1)andwhichwederiveforp2(1;2)bysettingkgkW1;p0CsupF2LqkFkLq1RrgF.SuchFcanbedecomposedinr'for'2W1;q0andsomedivergencefreeterm,andbytheestimatesforq�2wehavekr'kLqCkFkLq.13 Thenthereexistsome2W2;2(D2;som)withRD2=0andsomeP2W2;2(D2;SOm)withP�I2W1;20(D2;Rmm),whereIdenotestheidentitymatrix,suchthatr?=P�1rP+P�1 Ppointwisea.e.inD2.(A.3)Inaddition,wehavetheestimateskkW1;2(D2;Rmm)+kP�IkW1;2(D2;Rmm)Cmk kL2(D2;Rmm2)(A.4)andkkW2;2(D2;Rmm)+kP�IkW2;2(D2;Rmm)Cmk kW1;2(D2;Rmm2):(A.5)Inordertoprovethislemma,wei

17 ntroduceforyettobechosen"mandCmthesetU&#
ntroduceforyettobechosen"mandCmthesetUU"m;Cm=t2[0;1] Thereisadecompositionoft and(A.3){(A.5)hold.Thissetisclearlynon-emptyas02U(using0andPI).Furthermoreitisclosed,dueto(A.5).Toproveopennesswe xsomet02U,t01.Byde nitionofUwethen ndsomet02W2;2(D2;som R2)andRPt02W2;2(D2;SOm)suchthat(A.3),(A.4),(A.5)holdwhereandParereplacedbyandR,respectively.WenowprovethefollowingPropositionA.6De netheoperatorT:W2;2\W1;20(D2;som)W1;2(D2;som R2)!L2(D2;som);T(U;):=div(e�UreU+e�U(r?+)eU):Then,thereisaconstant &#x]TJ/;ø 9;&#x.962; Tf;&#x 11.;в ;� Td;&#x [00;0suchthatthefollowingholds:IfkrkL2(D2) istrue,thenthereexistssom

18 e &#x]TJ/;ø 9;&#x.962; Tf;&#x 11
e &#x]TJ/;ø 9;&#x.962; Tf;&#x 11.;в ;� Td;&#x [00;0suchthatforevery2W1;2(D2;som R2)withkkW1;2(D2;Rmm) we ndsomeU2W2;2\W1;20(D2;som)suchthatT(U;)=0:Furthermore,Udependscontinuouslyon.Proof.Firstofall,wenoticethatTiswellde nedandsmooth,astheexponen-tialfunctionmapsW2;2intoW2;2smoothly.Furthermore,wehaveT(0;0)=0.Thepropositionfollowsfromtheimplicitfunctiontheorem,ifwecanprovethatthelinearizationinthe rstcomponentofTat(U;)=(0;0),namelyH( ):=4 +r r?�r? r;isanisomorphismH:W2;2\W1;20(D2;som)!L2(D2;som):Theinjectivityfollowsforsmall&#x]TJ/;ø 9;&#x.962; Tf;&#x 11.;в ;� Td;&#x [00;0asin[Uhl82]:For1p2wehavekH( )kLpk4 kLp�Ckr kLpk

19 rkL2c0k kW2;p�k kW2;p ;
rkL2c0k kW2;p�k kW2;p ;14 [Pom92]C.Pommerenke.BoundaryBehaviourofConformalMaps.DieGrundlehrendermathematischenWissenschaften299.Springer-Verlag,Berlin-Heidelberg-NewYork,1992.[Qin93]J.Qing.Boundaryregularityofweaklyharmonicmapsfromsurfaces.J.Funct.Anal.,114:458{466,1993.[Riv07]T.Riviere.Conservationlawsforconformallyinvariantvariationalproblems.Invent.Math.,168(1):1{22,2007.[Riv08]T.Riviere.TheroleofIntegrabilitybyCompensationinConformalGeometricAnalysis.AnalyticaspectsofproblemsfromRiemannianGeometryS.M.F.,toappear,2008.[RS08]T.RiviereandM.Struwe.Partialregularityforharmonicmapsandrelatedproblems.Comm.PureAppl.Math.,61(4):451{463,2008.[Sem94]S.Semmes.AprimeronHardyspaces,andsomeremarksonatheoremofEvansandMuller.Co

20 mmun.PartialDi er.Equations,19(1-2):
mmun.PartialDi er.Equations,19(1-2):277{319,1994.[Ste93]E.M.Stein.Harmonicanalysis:Real-variablemethods,orthogo-nality,andoscillatoryintegrals.WiththeassistanceofTimothyS.Murphy,volume43ofPrincetonMathematicalSeries.PrincetonUniversityPress,Princeton,NJ,1993.[Str03]P.Strzelecki.AnewproofofregularityofweaksolutionsoftheH-surfaceequation.Calc.Var.PartialDi er.Equ.,16(3):227{242,2003.[Tar85]L.Tartar.RemarksonoscillationsandStokes'equation.InMacro-scopicmodellingofturbulent ows(Nice,1984),volume230ofLec-tureNotesinPhys.,pages24{31.Springer-Verlag,Berlin,1985.[Uhl82]K.K.Uhlenbeck.ConnectionswithLpboundsoncurvature.Com-mun.Math.Phys.,83(1):31{42,1982.[Wen69]H.C.Wente.Anexistencetheoremforsurfacesofconstantmeancurvature.J.Math.Anal.Appl.,26:318{3

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