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Minimization of Linear Growth Functionals of Measure

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Ithankmyadvisors,Dr.ChenandDr.Rao,fortheirsupportanddirectionthroughoutmystudies.IwouldliketothankDr.McCullough,Dr.KeeslingandDr.Kumarfortheiradviseandcounsel.ThanksgotoSujeetBhatforhisfriendshipandsupport,particularlyinthenalstagesofthisproject.Finally,Ithankmyfriendsandfamilyforalloftheirkindwordsofsupportandencouragement. iv

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page ACKNOWLEDGMENTS ............................. iv ABSTRACT .................................... vi CHAPTER 1INTRODUCTION .............................. 1 2CONVEXFUNCTIONSOFAMEASURE ................ 6 2.1Introduction. .............................. 6 2.2NotationandPreliminaries. ...................... 9 2.3MainResult. .............................. 12 3APARTIALREGULARITYRESULTFORPLASTICITY ....... 27 3.1Introduction ............................... 27 3.2DecayEstimate ............................. 30 3.3RegularityoftheElasticRegion .................... 60 4CONCLUSION ................................ 63 REFERENCES ................................... 65 BIOGRAPHICALSKETCH ............................ 68 v

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Therearemanyapplicationsthatinvolvetheminimizationofaconvex,lineargrowthfunctionofameasure.Forexample,imagerestorationmodels,Plateau'sproblemanddeformationofathinplate(theplasticityproblem)involveminimizingsuchfunctions.Inordertounderstandthetheoryoftheseproblems,wemustunderstandhowtogivemeaningF(),whereisavectorvaluedmeasureandFisaconvexfunctionwithlineargrowth. Inthisdissertation,weusethespaceofcontinuous,boundedfunctionstodenetheFencheltransformofafunctionofmeasure.Wethenshowthatunderthisdenition,thedoubleFencheltransformcoincideswiththedenitiongivenbyAnzellottiandGiaquintaandusedthroughouttheliterature.Thelowersemi-continuityofthefunctionalRF()isadirectresultofpropertiesoftheFencheltransform. Weusethisformulationtoestablishapartialregularityresultfortheelastic-plasticdeformationproblem.WeshowthatthedomainmaybedecomposedintoanopenelasticregionEandaclosedplasticregionP.OnE,thesolutionusatisestherelatedPoissonequationandisregular.Weuseadecay vi

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vii

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Therearemanyapplicationproblemsinvolvingvariationalintegralsoftheform minZF(x;u;Du);(1{1) foropenRn,whereu=(u1(x);:::;uN(x))isavectorvaluedfunctionandF(x;u;p)isconvexinp.Forexample,suchminimizationproblemsareusedinimagedenoisingandedgedetection,modelingthedeformationofathinplateanddeterminingasurfaceofminimalareawithprescribedboundaryconditions.Infact,Hilbert's19thand20thproblemsdealwiththese\regularproblemsinthecalculusofvariations,"forn=2andN=1;seeGiaquinta[ 12 ]foracomprehensiveoverviewofthesetypesofproblemsaswellasextensivereferences.In1912,Bernstein[ 4 ]usedthecalculusofvariationsmethodtoestablishexistenceandregularityresultsforthe2-dimensionalreal-valuedDirichletproblem.Serrin[ 25 ]appliedsimilarmethodstoextendtheseresultsforn-dimensions.Wewishtoexploretheminimizationproblemforu2BV(;RN);inthiscaseDuisaRadonmeasureandweneedgivemeaningtothevariationalintegral( 1{1 ). Forexample,Giusti[ 17 ]considersaminimalsurfaceareaproblem(Plateau'sProblem),where Inthiscase,ZF(Du)dx=Zq 1

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AnzellottiandGiaquinta[ 1 2 ],HardtandKinderlehrer[ 18 ]andZhou[ 28 ]studyanti-planarshearofathinplate(theplasticityproblem).Hereoneseekssolutionstotheminimizationproblem( 1{1 ),where 2jpj2ifjpj;jpj1 2ifjpj>;(1{3) forsomethreshold>0.Inthiscase,u:!Risthedisplacementoftheplate. ChambolleandLions[ 5 ]proposedamodeltorecoveranimage,u,fromanobservednoisyimageI=u+noisebyminZF(Du)+Z 1{3 ).ThediusionfromthisminimizationmodelisstrictlyperpendiculartothegradientwhenjDuj>,whereedgesarelikelytobepresent,andisotropicwhenjDuj.Thusthemodelpreservesedgesandeliminatesnoise. Additionally,Chen,LevineandRao[ 6 ]consideredafunctionofq(x)growth,forq(x)1,whichisusedasamodelforimagedenoising,enhancementandrestoration.Inthispapertheyproposedanewmodelforimagerestoration.TheproposedmodelincorporatesthestrengthsofthevarioustypesofdiusionarisingfromtheminimizationproblemminZjDujp+ 1{1 )with

PAGE 10

where>0isxedand1
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problemsdependsonhowonedenesZF(x;u;Du); 2 ]deriveaformulaforthespecialcaseoftheplasticityproblem,withFasin( 1{3 ).Wetakeadierentapproach;weusethedoubleFencheltransformtogetaformulaforthevariationalintegralZF(Du); Wewishtoestablishsomeregularityresultsforsolutionstofunctionalswithlineargrowth.Ingeneral,theminimumofaconvex,lineargrowthfunctionalmayhavesingularities,eveniftheintegrandissmooth[ 12 13 14 16 ].Inapplications,numericalresultshaveshowntheeectivenessofvariousconvex,lineargrowthfunctionalsinfeature-preservingimageprocessing.Imagesrestoredwiththesemodelsaresmootherinregionswherethegradientissmallandtheedgescorrespondtoplaceswherethegradientislarge|adesirablepropertyforimagedenoising. Inparticular,weshallfocusonaproblemthatarisesfromanti-planarshearforelastic-plasticmaterials, infv2AZF(Dv)Zfv;(1{6) wherev:Rn!R,theverticaldisplacementofmaterial,belongstoanappropriatesetAandfisagivenexternalforce.Fisdenedin( 1{3 ).NotethatFisaC1convexfunctionwithlineargrowth.TheminimizationproblemistakenoverthefunctionspaceBV(),witheitherDirichletorNeumannboundaryconditions.

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Fromnumericalresultsinimageprocessing(ordeformationofathinplate),weseethatthesolutionappearstobesmoothwhenthegradientissmallbutnotwhenthegradientislarge.Ourgoalistoverifythisobservationtheoretically.Thatistosay,supposeuisaminimizerofthefunctional( 1{6 ).WewishtodecomposeintosetsEandP,whereEistheelasticregion,denedLn-a.e.byE=fx2:jru(x)j<1g 18 ]forsimpleone-dimensionalexamples). WehavefollowedtheschemesetforthinanunpublishedworkbyTonegawa[ 26 ].Whilethegeneraloutlineoftheprooffollowsthiswork,wehaveprovidedmanymissingdetails.Inparticular,wehaveprovidedadetailedprooffortheestimatesoftheHoldernormofasolutiontoanauxiliaryproblem,establishedtherstvariationformula,andlledindetailsfortheproofsofthemainpropositions.Additionally,wehavecorrectedamistakeinthehandlingofanestimateinvolvingtheintegralontheboundarybyusingthetracetheoremforBVfunctions.

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14 ]denedafunctionF(x;p0;p):=Fx;p p0p0; existsunderappropriateconditions(givenbelow).Letu2BV()andchooseapositiveRadonmeasuresothatthetotalvariation,jDuj,andtheLebesguemeasure,Ln,areabsolutelycontinuouswithrespectto.DenotetheRadon-NikodymderivativesofLnandthevector-valuedmeasureDuwithrespecttobydLn d; dd: 22 ]tothisdenitiontoestablishthelowersemi-continuityoftheintegral. 6

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AnzellottiandGiaquinta[ 1 ]appliedthisdenitiontotheplasticityproblemwithFasin( 1{3 )with=1.Here,theauthorsdenedafunctiononRRnbyF(t;p):=8>><>>:Fp ttt>0;limt&0Fp ttt=0: djjdjj;(2{1) wherejjisthetotalvariationof,andd0 djjaretheRadon-Nikodymderivatives.InLemma1.1[ 2 ],theauthorsused( 2{1 )toshowthat foru2BV().HereDuisdecomposedintoitsabsolutelycontinuousandsingularpartswithrespecttoLebesguemeasure,i.e.,Du=rudx+Dsu: 2{2 )followsfrom( 2{1 ),wedecomposewithrespecttothemeasuresrudxandDsu.Thatistosay,sincethemeasuresaremutuallysingular,thereexistsasetAonwhichruisnotzeroandDsuisidenticallyzero.ThusonA,Du=rudxandonthecomplement,Du=Dsu.Splittingtheintegralin( 2{1 )intothesumofintegralsoverAandnAgives( 2{2 ). Inamoregeneralsetting,AnzellottiandGiaquinta[ 3 ]providedauniedapproachtothepartialregularitytosolutionsoftheminimizationproblem( 1{1 )forageneralconvexfunctionFwithgrowthm1;thatistosay,therearepositiveconstantsandsothat

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Inthiscase,theauthorsdened whereF1(p):=limt!11 2{1 )andthesetechniquestoestablish( 2{4 )inthegeneralcase.Thedenition( 2{4 )isusedbyDemengel,Hardt,Kinderlehrer,Temam,TonegawaandZhouamongothersthroughouttheliteratureinthestudyofexistenceandpartialregularityofsolutionstotheminimizationproblemwithFasin( 1{3 )[ 8 18 19 20 21 28 ]. Whilethedenition( 2{4 )issucientforthestudyoffunctionalsofthistype,therearestillunsettledquestions.Inparticular,whatmotivatesthisdenition,howdoesitrelatetopreviousresultsandwhatistherelationtoconvexanalysis?ThepurposeofthisdissertationistotakeaverydierentroutetothedenitionofF(m)whereFisaconvexfunctiononRnandmisavectormeasure.OurapproachistouseFencheltransformstodeneRF(m)in( 2{4 ).Briey,givenaconvexfunctionFonRn,considertheconvexfunctionalonL1(;Rn)denedbyf7!ZF(f)dx:

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Sinceboundedvector-valuedRadonmeasuresarelinearmapsonCB,wecannaturallyrepeatthisprocedureonthespaceofboundedvectormeasuresM,anddeneF(m):=sup2CBZdmF(): 2{4 )inthecontextofconvexanalysis.Oneimmediateconsequenceisthelowersemi-continuityofthisfunctional.Additionally,thistechniquemaybeappliedtodeneconvexfunctionalsofobjectsmoregeneralthanmeasures(e.g.,certaintypesofoperators).AlongthewaywealsondveryinterestingpropertiesofF.Justtonameone:F(m+n)=F(m)+F(n); forsome;;>0.DenoteM=fboundedRn-valuedmeasuresong:

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Foranym2M,letjmjdenotethetotalvariation.WemaydecomposemintoitsabsolutelycontinuousandsingularpartswithrespecttoLebesguemeasure:m=ma+ms: 7 ]. SupposeFsatisestheconditionsabove.Iff2L1(;Rn),thelineargrowthconditiononFgivesZ(jfj)dxZF(f)dxZ(jfj+)dx; i.e.,F2L1.TheFencheltransform,F,ofFisdenedonCBby Observethatforxedf2L1,themappingLisacontinuous,anemap.HenceF()isconvexandlowersemi-continuousonCB.Thusthedesiredlowersemi-continuityfollowsautomaticallyfromtheFencheltransformapproach.SeevanTiel[ 27 ]andEkelandandTemam[ 9 ]formoreonthegeneraltheoryofFencheltransforms. Weobservethat

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Form2M,denethedoubleFencheltransform ByProposition 2.1 ,wemaydene IfweendowCBwiththeuniformnorm,thenMCB.ThecoarsesttopologyonMsothatthemappingm7!RdmiscontinuousiscalledtheweaktopologyonM.By( 2{7 ),F(m)islowersemi-continuousintheweaktopologyandconvexonM. Foreachf2L1,fdx2M.Itiseasytosee 2{8 ),F(fdx)=sup2CBjj1ZfdxF();

PAGE 19

soZF(f)dxZfdxF(): OurgoalistoderiveanexplicitformulaforthedoubleFencheltransformF.Thiswehopewilljustifythedenitiongivenin( 2{4 ).

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Wepresenttheproofasaseriesofclaims: asjj1.Theclaimfollowsimmediatelyfrom( 2{8 ).

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SinceA(f)andB(f)aredisjointforeachf2L1,wehavebyclaim3thatF(1)+F(2)=supf2L1Z1(f1A)F(f1A)dx+supf2L1Z2(f1B)F(f1B)dx=supf2L1Z1(f1A)F(f1A)dx+Z2(f1B)F(f1B)dx=supf2L1Z1(f1A)+2(f1B)(F(f1A)+F(f1B))dx=supf2L1Z(1+2)(f1A[B)F(f1A[B)dx=F(1+2);

PAGE 23

2{8 )wehaveF(m):=supjj1ZdmF()supjj1spt()UZdmF():

PAGE 24

asjmj(Kc)=0.Therefore,wehaveF(m)=supjj1ZdmF()supjj1ZdmF()supjj1spt()UZdmF();

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=supjj1Z1dm+Z2dnF(1)+F(2)=supjj1Z1dm+Z2dnF(1+2)=supjj1Z1+2d(m+n)F(1+2)=supjj1spt()U[VZd(m+n)F()=F(m+n);

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Conversely,byclaim2wehaveF(m1Kj+n1Lj)=F(m+n)1Kj[LjF(m+n)+jm+nj((Kj[Lj)c)F(m+n)+jmj((Kj[Lj)c)+jnj((Kj[Lj)c): Therefore,byclaim7wehaveF(m+n)=limF(m1Kj+n1Lj)=limF(m1Kj)+F(n1Lj)=limF(m1Kj)+limF(n1Lj)=F(m)+F(n) asdesired.ThusProposition 2.3 isproved. Letmbeavectormeasure.Decomposemintoitsabsolutelycontinuouspart,ma,anditssingularpart,ms,withrespecttoLebesguemeasure.Sincema?ms,byPropositions 2.2 and 2.3 wehave djmjdjmj:

PAGE 27

djmj=(x0): 2.3 andProposition 2.2 ,wehaveF(m)liminfr!0F(x0)1B tFt(x0)=F1(x0)=ZF1(x)dx0=ZF1(x)djmj:

PAGE 28

Nowletx1;x2;:::;xkbeanitesetofdistinctpointsinand1;2;:::;k>0.Foreachi,denote(xi)=i2Sn1.Denotethemeasureiixi=mi.Wewillrefertoameasureoftheformm=kXi=1ii(xi)xi=kXi=1mi 2.3 and( 2{11 ),wehaveF(m)=kXi=1F(mi)kXi=1ZF1(i)djmij=ZF1dm djmjdjmj: djmjdjmj: djmjdjmj=limi!1ZF1dmi

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Let1;2;:::beasequenceintheunitballoftheBanachspaceC(;Rn)withthesupremum.Thenforanyandanysequencen2M,wedenen*weaklyifandonlyiflimn!1hj;ni=hj;i; ByLusin'sTheorem,foreachkwemaychooseacompactsetCksuchthatjmj(Cck)1

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From( 2{10 )andthepreviousproposition,wehave Indeed,letm2MandsupposethatGislowersemi-continuousandconvexonMsuchthatG(fdx)RF(f)dx.LetG(m).ThenbytheHahn-BanachTheoremthereexists2CBandanumber>0suchthatforalln2M,G(n)Zdn;and
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Takingthesupremumoverf2L1,weconcludethatF().Thusfor2CBchosenaboveandalln2MwehaveF(n)=supjj1ZdnF()ZdnF()Zdn: Finally,itremainstoshowthat 2.5 ,weneedonlyshowthattherighthandsideisconvex.SinceFisconvex,weknowthatF1isconvexaswell.Fortheconvenienceofthereader,weshowtheconvexityofthemap Theabsolutelycontinuous(singular)partofasumofmeasuresisthesumoftheirabsolutelycontinuous(singular)parts.Thusweneedonlyshowtheconvexity

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of Tothisend,letm;n2Mbemeasuresandlet0t1;denotes=1t.Forbrevity,denoteA=d(tm+sn) tdjmj+sdjnj(tdjmj+sdjnj)ZtF1dm tdjmj+sdjnj+sF1dn tdjmj+sdjnj(tdjmj+sdjnj)(bytheconvexityofF1)=tZF1dm djmjC(tdjmj+sdjnj)+sZF1dn djnjD(tdjmj+sdjnj)

PAGE 33

=tZF1dm djmjC(tdjmj+sdjnj)+sZF1dn djnjD(tdjmj+sdjnj)=tZF1dm djmjdjmj+sZF1dn djnjdjnj; 2{15 )isconvex. Theequality( 2{13 )ispreciselytheresultwehavesetouttoestablish,andthusTheorem( 2.1 )isproved.

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1{6 );forsimplicity,wewilltake=1.LetRnbeaboundeddomainwithLipschitzboundary,andfunctionsf2L1()and'2L1(@)begiven.Foru2BV(),wedecomposeitsgradientmeasureDuintoitsabsolutelycontinuousandsingularpartswithrespecttoLebesguemeasure:Du=rudx+Dsu:Fromourpreviouswork,wemaydeneZF(Du)=ZF(ru)dx+ZjDsuj; 18 ]fortheDirichletproblem.ThereexistsaminimizerifkfkL1cD()1,wherecDisthesmallestconstantsuchthatkkL1()cD()ZjDj;

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forall2BV()withRdx=0.Zhou[ 28 ]studiedtheparabolicproblemassociatedwiththeabovefunctional.Hardt,TonegawaandZhou[ 20 21 ]studytherelatedgradientow,whereFisconvexandhaslineargrowth. Below,weshowthefollowingregularityresultatapointwherethesolutionisclosetoalinearfunctionwithslopestrictlysmallerthanone: 1{6 )witheitherDirichletorNeumannboundarycondition.Foranygiven0<<1,thereexistpositiveconstants0and0,whichdependonlyonnandsuchthat,if1 1{6 )withDirichletorNeumannboundarycondition.IfLn(fjruj<1g)>0,thenthereexistsanonemptyopenelasticregionEonwhichuisinC1;,jruj<1andusatisesu=fonE:

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Wenotethattheprevioustworesultsdependonlyontheminimizeruandnotontheinitialorboundaryconditions.However,itmayhappenthattheLn-measureofthesetfjruj<1giszeroforboththeDirichletandNeumannproblems.Thefollowingconditionsontheboundaryvalue'andtheforcetermf,whicharephysicallyreasonable,assurethatthethereexistsanonempty,openelasticregion. 3.2 3.2 3.1 isthedecayestimategiveninProposition 3.3 below.Weshowthatthefunctional,denedbelow( 3{7 ),whichisanaverageofthegradientmeasure,decaysforsmallballswithalinearcorrection.Weachievethedecaybyapproximatingaminimizerto( 1{6 )byaLipschitzfunctionandestablishinganL1estimatefortheirdierence.ThelinearcorrectionarisesbyapproximatinguwiththesolutionofanappropriatePDE.

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1{6 )forsomefunctionuofboundedvariationintermsafunctionwithgradientstrictlysmallerthanone. 2ZBr\fjruj<1gjr(uh)j2dx+ZBrjDsuj+ZBrDsurh 2+1 2jrhj2rurh1 2(2jruj1jrhj)(1jrhj)1 2jruj2=2 2jruj21 2jrhj2r(uh)rh=1 2jr(uh)j2:

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sincejrhj1.Hence,bycombiningthethreeinequalitiesabovethedesiredestimateisdeduced. Letf2L1()begiven,andxB2r(a).Alsoletv2C0;1(B2r(a))begiven.Below,weusethesmoothingofLipschitzfunctionsadoptedfromSchoenandSimon[ 24 ].For>0and>0,whichwewillchooselater,assumethatthereexistsl2RnsuchthatsupB2r(a)jrvlj2andjlj12: 24 ].

PAGE 39

Forthesecondinequality,weobservethatv(x)v(x)=(rv)(x)v(x)=Zjxyjrr(xy)[v(y)lx]dy(v(x)lx)=v(x)v(x): Furthermore,bythemeanvaluetheorem,thereisa2Br(x)sothatjv(x)v(x)j=Zjxyjrr(yx)[v(y)v(x)]dyZjxyjrr(yx)jrv()jjyxjdy(r)supBr(a)jrvj2r=r1+2;

PAGE 40

Finally,forthelastestimate,letx6=y2Br(a).Thenwehavejrv(x)rv(y)j jxyj=jrv(x)rv(y)j jxyj=jxyjZRnr(xz)r(yz)(rv(z))dzsupBr(a)jrvjjxyjZRnjr(xz)r(yz)jdz=supBr(a)jrvjjxyj(r)nZRnxz ryz rdz=supBr(a)jrvj(r)nZRnxz ryz r jx0y0jc32r=c3r: FromtheusualtheoryofPoissonequations(seeGilbargandTrudinger[ 15 ],forexample),forany~r2[r Moreover,wehavethefollowingestimates. 3{1 ).Then(i)rsupB~r(a)jrw(x)rw(y)j jxyjc4rkfkL1+;(ii)supB~r=2(a)jrw(x)rw(y)j jxyj1=2c51 2Z@B~r(a)jvjdS+r1=2kfkL1:

PAGE 41

3{1 )andwritew=w1+w2,wherewehavew1;w22W1;2(B~r(a))\C1;(B~r(a))sothat8>><>>:w1=fonB~r(a);w1=0on@B~r(a);and8>><>>:w2=0onB~r(a);w2=von@B~r(a): jxyjckfkL1r1: where[];BrdenotestheHoldersemi-normontheballofradiusrwithexponent[ 15 ,Theorem8.33]. ItremainstoestimatetherighthandsideonlyintermsoftheHoldernorm.Wehave[ 15 ,Theorem8.16],k~wkL1(Br)sup@Br~w+ckrvkL1(Br)=ckrvkL1(Br):

PAGE 42

Replacevby~v=vl(xa),wherel=rv(y),forsomexedy2B~r(a).Notethat~v=vand[r~v];Br=[rv];Br.WeclaimthatkrvlkL1(Br)cr[rv];Br: jxyj: jxyj=cr[rv];Br: 3.2 .Itfollowsimmediatelythat[Dw2];Brcr: jxyjsupB~r(a)"jrw1(x)rw1(y)j jxyj+jrw2(x)rw2(y)j jxyj#c4r1kfkL1+r For(ii),wenotethatfor=1=2,therstpartgives[rw1]1=2;B~r=2[rw1]1=2;B~rckfkL1(B~r)r1=2:

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Forw2,weuseGreen'srepresentationw2(x)=Z@Br(0)K(x;y)v(y)dSy=Z@Br(0)r2jxj2 jx~xj1=2Z@BrjrKxi(;y)jjx~xj1=2jv(y)jdSyc !nrn+1jx~xj1=2Z@Brjv(y)jdSyc rn+1 2Z@BrjvjdSy: jxyj1=2jrw1(x)rw1(y)j jxyj1=2+jrw2(x)rw2(y)j jxyj1=2c5r1=2kfkL1+1 2Z@BrjvjdS 3.1 thatsupB~r(a)jrwljc+rkfkL1:

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Indeed,forx2B~r(a)andy2@B~r(a)wehavejrw(x)ljjrw(x)rv(y)j+jrv(y)lj(2r)supx;y2B~r(a)jrw(x)rw(y)j jxyj+jrv(y)ljc+rkfkL1; 3.2 toestimatejrv(y)lj2. @ndS+ZB~r(uw)fdx+ZB~rjDsuj+2 2ZB~r\fjruj<1gjr(uw)j2dx:

PAGE 45

WemaynowapplyLemma 3.1 withwinplaceofhtogetZB~rF(Du)ZB~rF(rw)dxZB~rr(uw)rwdx+ZB~rDsurw+ZB~rjDsuj+2 2ZB~r\fjruj<1gjr(uw)j2dx: @ndSZB~r(uw)fdx; Afunctionu2BV()issaidtobealocalsolutioninprovidedthat forany2BV0().ObservethataminimizerfortheDirichletortheNeumannproblemisalocalsolution.Wenotethatthefollowingargumentrequiresonlythatubealocalsolution. Next,weestablisharstvariationformulabycomputingtheEuler{Lagrangeequationforthefunctional( 1{6 ).

PAGE 46

NotethatbyDirichlet'sprinciple(seeEvans[ 10 ]),itnowfollowsthatusatisesu=f,whenjruj1anduisfreeofsingularpart.

PAGE 47

3{1 )andsupposethatuisalocalsolution.For~r2[r 11 ],forexample)wehavew2BV().Thusfromthedenitionoflocalsolution,weseethat sinceDw=rw.SinceRB~rF(Du)+R@B~rjDsujRB~rF(Du),( 3{3 )maybereducedtoZB~rF(Du)ZB~rF(rw)ZB~rf(uw)dxZ@B~rF(Dw): 11 ],wehavethatDw=(TwTu)dHn1on@B~r;

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3.2 .Ifu2BV()isalocalsolutionandwisasolutionto( 3{1 ),thenZB~rjDsuj+ZB~r\fjruj1gjrujdx+ZB~r\fjruj<1gjr(uw)j2dxc8Z@B~rjuvjdS+c9rn1+2: 3.4 and 3.3 ,wehaveZ@B~rjuvjdS+ZB~rf(uw)dxZB~rF(Du)ZB~rF(rw)Z@B~r(uv)@w @ndS+ZB~rf(uw)dx+ZB~rjDsuj+2 2ZB~r\fjruj<1gjr(uw)j2dx: 2ZB~r\fjruj<1gjr(uw)j2dxZ@B~rjuvjdS+Z@B~rjuvj@w @ndS=Z@B~rjuvj1+@w @ndS+Z@B~rjvvj1+@w @ndS

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3.1 gives@w @njlj+c4+rkfkL1candwehavejvvjr1+2byLemma 3.2 .Thedesiredinequalityfollowsimmediately. BelowweshowthatifuisalocalsolutionandvisaLipschitzfunctionwithsmallgradientthatcoincideexceptforasetofsmallmeasure,thenwecanestimatekuvkL1.TheresultisamodicationofHardtandKinderlehrer([ 19 ,Theorem2.2]),whichwepresentheretoillustratetheimportantchanges. 2jBj;forallr2r. Thenthereexitspositiveconstantsc10andc11sothatifLn(fu6=vg\B2r(a))c10rn;

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denelater.Supposethat0<>>>>><>>>>>>:1inB;(h)1(hjxj)inBhnB;0innBh: Observethatbyconvexity,F(Dv)=F(rv)+jDsvjFp(rv)rv+jDsvj=Dv; 2{5 ),thereexistscsothatjpjc(F(p)+1).RecallthatjDvj=jrvj1.Wehavethefollowingestimate:ZBhjD[(uv)]jZBh0(uv)jDuDvjZBh0(uv)jDuj+ZBh0(uv)jDvjcZBh0(uv)F(Du)+c0ZBh0(uv)ZBhD[(uv)]+ZBhj0(uv)Dvj+cZBh0(uv)ZBhD[(uv)]+cZBh0(uv):

PAGE 51

Substituting( 3{4 )intotheaboveinequality,weseethatZBhjD[(uv)]j(h)1ZBhnBj(uv)j+ZBhj(uv)fj+cZBh0(uv)(h)1ZBhnBj(uv)j+kfkL1ZBhj(uv)j+cjspt((uv))j; asspt()=Bh,and0(t)1. Let0>>>>><>>>>>>:0fortk;tkfork
PAGE 52

FromthedenitionsofandA(k;h),andtheisoperimetricinequality(seeEvansandGariepy[ 11 ,Theorem5.6.1,partiii])wehave(sk)jA(s;)jn1 n1dx!n1 forrh2rand0
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ButA(d;r)=fjuvj>dg\Br,sojA(d;r)j=0impliesthatjuvjdonBr.ThatistosaythatkuvkL1(Br)d=c11(Ln(fu6=vg\B2r))1=n 3.6 totheproposition,wetakec10=sotheconditionLn(fu6=vg\B2r)=jA(0;2r)jrn Inthefollowingproposition,wedenote (r;l;x):=1 forr>0,l2Rnandx2. WeshowthatonecanndaLipschitzfunctionvthatapproximatesalocalsolutionu.Choosinga\goodslice"ofballandassumingthatissmall,weshowthatdecaysforasmallerball,withtheadditionofasmalllinearcorrection.FortheproofofthestandardLipschitzapproximation(steps1and2,below),wereproducetheproofofTheorem2insection6.6.2[ 11 ]withsomemodicationtosuitourneeds.

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2(4r;l1;a)+c18r2kfk2L1: 1. DeneasetRwhere(;l1;x)issmallandestimateLnB2r(a)nR. 2. Letg(x)=u(x)l1xandshowthatgisLipschitzonR.Usingastandardextensiontheorem,weestablishtheexistenceofaLipschitzfunctionvonB2rsuchthatv=uLn-a.e.onR. 3. Usestep 1 toestimatethesizeoffu6=vg. 4. EstimateR@B~rjuvjdSintermsof(4r;l1;a)andapplyLemma 3.5 5. Assembletheabovepiecestogetthedesiredestimateon(!r;l2;a). 6. Notethatintheprocessoftheproof,wehavewhatweneedtoobtainthedesiredestimateforjl1l2j.

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Thus,wecanestimateLnB2r(a)nR1Xi=1jB5ri(xi)j=5n1Xi=1jBri(xi)j5n n1(B)c0 nZjDgj=c nZjDul1jc22(;l1;x)1=2:(3{8)

PAGE 56

Itnowfollowsthat(g)x;=2k+1(g)x;=2k=1 jB(x)\B(y)j"1

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forLn-a.e.x;y2RB2r(a).Dene=c2264,sothatforLn-a.e.x;y2Rwehaveju(x)u(y)j=g(x)+l1xg(y)+l1yjg(x)g(y)j+jl1(xy)j2+jl1jjxyj: 11 ]),wemayextendutoaLipschitzmappingv:B2r(a)!Rsuchthatv=uLn-a.e.onRandsupB2r(a)jrvl1j=supB2r(a)jrgj2: 8(n+1); Step4.Weneedthefollowing

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10r: 10randjBj3 10r: Forthischoiceof~r,wehave

PAGE 59

ForrkfkL1c7and(4r;l1;a)c28(thelatterbeingsmall),Proposition 3.2 andtheestimate( 3{9 ),wehave5 r(Ln(B2r(a)\fu6=vg))1=nLn(Br(a)\fu6=vg)c0 2n; 2nbyourchoiceofabove.Thus,from( 3{10 ),weconcludethat 2n:(3{11) ByLemma 3.5 andtheprecedinginequality,wehaveZB!r(a)jDsuj+ZB!r(a)\fjruj1gjrujdx+ZB!r(a)\fjruj<1gjr(uw)j2dxc8Z@B~r(a)juvjdS+c9rn1+2c8c29rn(4r;l1;a)1+1 2n+c9rn(4r;l1;a)1+1 4(n+1)c30rnh(4r;l1;a)1+1 2n+(4r;l1;a)1+1 4(n+1)i; Step5.Notethatja+bj22jaj2+jbj2foranya;b2R.Foranyl22R,wecanestimateZB!r(a)\fjruj<1gjrul2j2dx2ZB!r(a)\fjruj<1gjr(uw)j2+jrwl2j2dx;

PAGE 60

Usingthelasttwoestimatesabove,wehave 2n+(4r;l1;a)1+1 4(n+1)i+c36ZB!r(a)jrwl2j2dxc35rn(4r;l1;a)1+1 4(n+1)+c36ZB!r(a)jrwl2j2dx:(3{12) WenowuseProposition 3.1 forthegradientofwtoestimatethelasttermabove.Let~w(x)=w(x)(u)a;rl1(xa)andnotethefollowing:8>><>>:~w=w=fonB~r(a),~w=v(u)a;rl1(xa)on@B~r(a), for~r2[r 3.1 to~w,yieldingsupx;y2Br=4(a)jrw(x)rw(y)j jxyj1=2supx;y2B~r(a)jr~w(x)r~w(y)j jxyj1=2c51

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Indeed,weestimateasfollows.1 3.2 ,wemayestimateI1byI1c~rn1~r1+2c~rn(4r;l1;a)1+1 4(n+1): 3{11 )thatI2:=Z@B~r(a)juvjdSc29rn(4r;l1;a)1+1 2n: 4(n+1)+rn(4r;l1;a)1+1 2n+ZBr(a)jDul1jc(4r;l1;a)1+1 4(n+1)+2nc(4r;l1;a)1+1 2n+2nc rnZBr(a)jDul1jc32(4r;l1;a)+c rnZBr(a)jDul1j: 3{8 )above,wehavec rnZBr(a)jDul1jc0(r;l1;a)1=2:

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Whence,1 rnZBr(a)jDul1jc32(4r;l1;a)+c0(r;l1;a)1=2c34(4r;l1;a)1=2 SinceB~r=2jBrj,itfollowsthatsupx;y2Br=4(a)jrw(x)rw(y)j jxyj1=2c51 4,wemaynowestablishadesirableestimateforZB!r(a)jrwl2j2dx=ZB!r(a)jrw(x)rw(a)j2dxcZB!r(a)jxajr1(4r;l1;a)+rkfk2L1+2(4r;l1;a)1=2kfkL1dxc(r!)n!(4r;l1;a)+r2!kfk2L1+2r!(4r;l1;a)1=2kfkL1c(r!)n!(4r;l1;a)+r2!kfk2L1+!(4r;l1;a)+4r2!kfk2L1c39(r!)n!(4r;l1;a)+r2kfk2L1; 3{12 ),wenowseethatjB!rj(!r;l2;a)c35rn(4r;l1;a)1+1 4(n+1)+c36ZB!r(a)jrwl2j2dxc35rn(4r;l1;a)1+1 4(n+1)+c(r!)n!(4r;l1;a)+r2kfk2L1:

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DividingbyjB!rj=c(r!)n,weconcludethat(!r;l2;a)c40!n(4r;l1;a)1+1 4(n+1)+c41!(4r;l1;a)+c42r2kfk2L1: 4(n+1)<1=4,wehavethedesireddecayestimate(!r;l2;a)1 4(4r;l1;a)+1 4(4r;l1;a)+c42r2kfk2L1=1 2(4r;l1;a)+c42r2kfk2L1: andnotethat8>><>>:h=w=finB~r(a);h=v(u)a;rl1(xa)on@B~r(a): Fromthegradientestimateforharmonicfunctions,wehaverh1(x)=1 15 ].Thusjrh1(x)j1

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Moreover,forh2,wehavethatjrh2jc31~rkfkL1: Theorem 3.1 isnowprovedbyiteratingthedecayestimate. 3.1 )WewilluseProposition 3.3 iteratively.Toinitializetheinductiveargumentassumethat1 3.3 maybeapplied.Forthelastinequality,wecheckeachtermasfollows:Forthepartwithjruj<1,wehavejrul1j2=jrul1jjrul1jjrul1j(jruj+jl1j)2jrul1j:

PAGE 65

Forthepartwithjruj1,itsucestoshowthatthereexistsac>0sothatjrujcjrul1j.Infactjruj jrul1jjruj jrujjl1jjruj jruj(14); tconstantisdecreasing.ThuswehaveZBr(a)\fjruj<1gjrul1j2+ZBr(a)\fjruj1gjruj+ZBrjDsuj2ZBr(a)\fjruj<1gjrul1j+cZBr(a)\fjruj1gjrul1j+ZBrjDsuj=cZBrjDul1j Fortheinductivestep,choose0sothatc430andrestrictrsothatc18r2kfk2L1=2.Furthermore,weassumethatjlj1j12and 2j1r 2i1!ji1c44r2kfk2L1;(3{13) forj=2;:::;k.Weneedtoshow! 3.3 andcontinuetheinductivestep.Taking!1=4,wehaveforallkk1Xi=11 2i1!ki11 2k2(k1)c451 2k=2; 3{13 )! 2k1r 2k=2c44r2kfk2L1:

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ByProposition 3.3 andtheinductiveassumptionweseethatjlkjk1Xj=1jlj+1ljj+jl1jk1Xj=1"c19! 2j1r 2j=2c44r2kfk2L1#1=2+c20rkfkL1k1Xj=1! 2j1 2r 2j=4c1=244rkfkL1#+c20rkfkL1k1Xj=1! 3.3 toshow! 2kr 2(k+1)=2r2kfkL1; Whencelimk!1! uniformlyforallx2Br=2(a).Hence lim!01

PAGE 67

ItfollowsfromLemma1insection1.6[ 11 ]thatjDsujBr=2(a)=0: weconcludethatLnBr=2(a)\fjruj1g=0: 3.2 .Assumethatuisaminimizerandthattheset~E=fjruj<1ghaspositiveLebesguemeasure. 3.2 )Sinceu2BV,itsderivativeDuisaRadonmeasurethatcanbedecomposedintosingularandabsolutelycontinuouspartswithrespecttoLebesguemeasure,Dsuandru,respectively.ItfollowsbyProposition6.8[ 23 ]thatjDsujissingularwithrespecttoLebesguemeasureaswell.Thus,byTheorem7.13[ 23 ],wehavefrom( 3{14 )limr!01 From~E,removepointsatwhicheitheroftheabovefailstohold;calltheresultingsetE.ThuswehaveLn(~EnE)=0,jruj<1onEandboththeabovelimitsholdforeachpointinE.Foreachxedx2E,thereexistssomex>0such

PAGE 68

thatjruj<12x: 3.1 givesthatjDsuj(Brx(x))=0andjruj<1xonBrx(x); Evenwiththesafe-loadcondition(whichguaranteestheexistenceofaminimizer),itisnotclearwhetherthesetEhaspositivemeasureornot.WenextproveTheorems 3.3 and 3.4 ,whicharesimplecomparisonarguments,discussedforexamplebyHardtandKinderlehrer[ 18 ]. 3.3 and 3.4 )Togetacontradiction,assumethatLn(fjruj<1g)=0: 2,a.e.Foragivenboundarydata'wecanchoosesomev2BV()withvj@='suchthatkvkBV()c()k'kL1(@)

PAGE 69

bythetracetheoremforBVfunctions([ 16 ]).Bytheminimalityofu,wehaveZF(Du)ZF(Dv)Zf(vu)dxc()k'kL1(@)+cD()kfkL1Z(jDujjDvj); 2ZjDujZjDuj1 2=ZF(Du)c()k'kL1(@)+1 3.2 applies.TheproofofTheorem 3.4 followssimilarlyusingv=0asacomparison.

PAGE 70

ThemainresultofChapter2,Theorem 2.1 ,showsthatthevariationalintegralRF(m),foravector-valuedmeasurem,canbedened\naturally"bytheFencheltransform.Animportantconsequenceofthetechniqueisthatthelowersemi-continuityofthefunctionalfollowsimmediatelyfromwell-knownpropertiesoftheFencheltransformfromcomplexanalysis.Inourapproach,wedecomposedthemeasurewithrespecttoLebesguemeasure:m=ma+ms,wheremaisabsolutelycontinuoustoLebesguemeasureandmsismutuallysingulartoLebesguemeasure.OnequestionofinterestthatwasnotexplorediswhetherLebesguemeasureisthebestchoiceforthebaseofthisdecomposition. ToestablishthedualityneededfortheFencheltransform,weconsideredthespaceofcontinuousboundedfunctions,CB,whichinducesatopology(theweaktopology)ontothespaceofboundedvector-valuedmeasures,M.IsCBthebestchoicetoestablishtherequiredduality?AnotherpossiblecandidateistheSobolevspaceW1;1(;Rn);withthischoice,theweaktopologyonMismetrizable. InChapter3,weestablishedapartialregularityresultfortheplasticityproblem.Thebasisofourtechniqueisthedecayestimateforthe\excess,"giveninProposition 3.3 .HardtandTonegawa[ 20 ]giveapartialregularityresultforaweaksolutiontotheevolutionproblem@u @t=divxFp(ru); 3.1 ;however,thedecaymethodtheyusedisdependentuponthespace 63

PAGE 71

variablehavingdimensionn=1or2[ 20 ].Oneextensionoftheirresultthattoconsideristoestablishthisresultforthegeneraln-dimensionalspacevariable. Anotherextensiontoexploreistoobtainapartialregularityresultfortheproblem@u @t=divxFp(ru)+1 2(uI); 2(uI(x))2(4{1) overBV().Animportantapplicationforthisproblemisimagerestoration.InthiscaseFmaybethefunctiongivenin( 1{3 )andIistheobservedimage.ThemaindistinctionbetweenthisproblemandtheoneconsideredbyHardtandTonegawa[ 20 ]isthedependenceof( 4{1 )onthefunctionuandthespacevariablex.

PAGE 72

[1] G.AnzellottiandM.Giaquinta.ExistenceofthedisplacementeldforanelastoplasticbodysubjecttoHencky'slawandvonMisesyieldcondition.ManuscriptaMath.,32(1-2):101{136,1980. [2] G.AnzellottiandM.Giaquinta.Ontheexistenceoftheeldsofstressesanddisplacementsforanelasto-perfectlyplasticbodyinstaticequilibrium.J.Math.PuresAppl.(9),61(3):219{244(1983),1982. [3] G.AnzellottiandM.Giaquinta.Convexfunctionalsandpartialregularity.Arch.RationalMech.Anal.,102(3):243{272,1988. [4] S.Bernstein.Surlesequationsducalculdesvariations.Ann.Sci.EcoleNorm.Sup.(3),29:431{485,1912. [5] A.ChambolleandP.-L.Lions.Imagerecoveryviatotalvariationminimizationandrelatedproblems.Numer.Math.,76(2):167{188,1997. [6] Y.Chen,S.Levine,andM.Rao.Variableexponent,lineargrowthfunctionalsinimagerestoration.SIAMJournalonAppliedMathematics,Inpress. [7] J.B.Conway.Acourseinfunctionalanalysis,volume96ofGraduateTextsinMathematics.Springer-Verlag,NewYork,secondedition,1990. [8] F.DemengelandR.Temam.Convexfunctionsofameasureandapplications.IndianaUniv.Math.J.,33(5):673{709,1984. [9] I.EkelandandR.Temam.Convexanalysisandvariationalproblems,volume28ofClassicsinAppliedMathematics.SocietyforIndustrialandAppliedMathematics(SIAM),Philadelphia,PA,1999.TranslatedfromtheFrench. [10] L.C.Evans.Partialdierentialequations,volume19ofGraduateStudiesinMathematics.AmericanMathematicalSociety,Providence,RI,1998. [11] L.C.EvansandR.F.Gariepy.Measuretheoryandnepropertiesoffunc-tions.StudiesinAdvancedMathematics.CRCPress,BocaRaton,FL,1992. [12] M.Giaquinta.Multipleintegralsinthecalculusofvariationsandnonlinearellipticsystems,volume105ofAnnalsofMathematicsStudies.PrincetonUniversityPress,Princeton,NJ,1983. 65

PAGE 73

[13] M.Giaquinta.Theproblemoftheregularityofminimizers.InProceedingsoftheInternationalCongressofMathematicians,Vol.1,2(Berkeley,Calif.,1986),pages1072{1083,Providence,RI,1987.Amer.Math.Soc. [14] M.Giaquinta,G.Modica,andJ.Soucek.Functionalswithlineargrowthinthecalculusofvariations.I,II.Comment.Math.Univ.Carolin.,20(1):143{156,157{172,1979. [15] D.GilbargandN.S.Trudinger.Ellipticpartialdierentialequationsofsecondorder.ClassicsinMathematics.Springer-Verlag,Berlin,2001.Reprintofthe1998edition. [16] E.Giusti.Boundaryvalueproblemsfornon-parametricsurfacesofprescribedmeancurvature.Ann.ScuolaNorm.Sup.PisaCl.Sci.(4),3(3):501{548,1976. [17] E.Giusti.Minimalsurfacesandfunctionsofboundedvariation,volume80ofMonographsinMathematics.BirkhauserVerlag,Basel,1984. [18] R.HardtandD.Kinderlehrer.Elasticplasticdeformation.Appl.Math.Optim.,10(3):203{246,1983. [19] R.HardtandD.Kinderlehrer.Variationalprincipleswithlineargrowth.InPartialdierentialequationsandthecalculusofvariations,Vol.II,volume2ofProgr.NonlinearDierentialEquationsAppl.,pages633{659.BirkhauserBoston,Boston,MA,1989. [20] R.HardtandY.Tonegawa.Partialregularityforevolutionproblemswithdiscontinuity.ManuscriptaMath.,90(1):85{103,1996. [21] R.HardtandX.Zhou.Anevolutionproblemforlineargrowthfunctionals.Comm.PartialDierentialEquations,19(11-12):1879{1907,1994. [22] Ju.G.Resetnjak.Theweakconvergenceofcompletelyadditivevector-valuedsetfunctions.Sibirsk.Mat.Z.,9:1386{1394,1968. [23] W.Rudin.Realandcomplexanalysis.McGraw-HillBookCo.,NewYork,thirdedition,1987. [24] R.SchoenandL.Simon.Anewproofoftheregularitytheoremforrectiablecurrentswhichminimizeparametricellipticfunctionals.IndianaUniv.Math.J.,31(3):415{434,1982. [25] J.Serrin.TheproblemofDirichletforquasilinearellipticdierentialequationswithmanyindependentvariables.Philos.Trans.Roy.Soc.LondonSer.A,264:413{496,1969. [26] Y.Tonegawa.Aregularityresultforplasticity.Preprint,1994.HokkaidoUniversity,Japan.

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[27] J.vanTiel.Convexanalysis:Anintroductorytext.JohnWiley&SonsInc.,NewYork,1984. [28] X.Zhou.Anevolutionproblemforplasticantiplanarshear.Appl.Math.Optim.,25(3):263{285,1992.

PAGE 75

IreceivedaBachelorofArtsdegreein1994fromOhioWesleyanUniversity.AtOWU,Imajoredinmathematicsandearnedminorsineconomicmanagementandphilosophy.In1997,IreceivedaMasterofArtsdegreefromBowlingGreenStateUniversityinmathematics.IenteredthedoctoralprogramattheUniversityofFloridain1998;IbeganworkwithDr.Chenin2001. 68


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MINIMIZATION OF LINEAR GROWTH FUNCTIONALS OF MEASURE


By

J. CHRISTOPHER TWEDDLE

















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

J. Cl'!I .-IIl 1. ir Tweddle


































I dedicate this work to my friends and family















ACKNOWLEDGMENTS

I thank my advisors, Dr. C'!i, i and Dr. Rao, for their support and direction

throughout my studies. I would like to thank Dr. McCullough, Dr. Keesling

and Dr. Kumar for their advise and counsel. Thanks go to Sujeet Bhat for

his friendship and support, particularly in the final stages of this project.

Finally, I thank my friends and family for all of their kind words of support

and encouragement.















TABLE OF CONTENTS


page


ACKNOWLEDGMENTS ............

ABSTRACT ....................

CHAPTER

1 INTRODUCTION .............

2 CONVEX FUNCTIONS OF A MEASURE

2.1 Introduction ... ............
2.2 Notation and Preliminaries.......
2.3 Main Result ... ............

3 A PARTIAL REGULARITY RESULT FOR

3.1 Introduction ...............
3.2 Decay Estimate .............
3.3 Regularity of the Elastic Region ....

4 CONCLUSION ................

REFERENCES ...................

BIOGRAPHICAL SKETCH ............


PLASTICITY















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MINIMIZATION OF LINEAR GROWTH FUNCTIONALS OF MEASURE

By

J. ('!0-Inlp!. r Tweddle

i ,v 2006

C'!i irw: Yunmei ('C1, i
Cochair: Murali Rao
Major Department: Mathematics

There are many applications that involve the minimization of a convex,

linear growth function of a measure. For example, image restoration models,

Plateau's problem and deformation of a thin plate (the plasticity problem) involve

minimizing such functions. In order to understand the theory of these problems, we

must understand how to give meaning F(p), where p is a vector valued measure

and F is a convex function with linear growth.

In this dissertation, we use the space of continuous, bounded functions to

define the Fenchel transform of a function of measure. We then show that under

this definition, the double Fenchel transform coincides with the definition given

by Anzellotti and Giaquinta and used throughout the literature. The lower

semi-continuity of the functional f F(p) is a direct result of properties of the

Fenchel transform.

We use this formulation to establish a partial regularity result for the

elastic-plastic deformation problem. We show that the domain 2 may be

decomposed into an open elastic region E and a closed plastic region P. On E,

the solution u satisfies the related Poisson equation and is regular. We use a decay









estimate to establish the desired regularity on a ball. Finally, we show that the

elastic region is nonempty for small load.














CHAPTER 1
INTRODUCTION

There are many application problems involving variational integrals of the

form

min F(x, u, Du), (1-1)
Jo
for open Q C R", where u (u'(x),.. u(x)) is a vector valued function and

F(x, u,p) is convex in p. For example, such minimization problems are used in

image denoising and edge detection, modeling the deformation of a thin plate and

determining a surface of minimal area with prescribed boundary conditions. In

fact, Hilbert's 19th and 20th problems deal with these -;,1! i r problems in the

calculus of variations," for n = 2 and N = 1; see Giaquinta [12] for a comprehensive

overview of these types of problems as well as extensive references. In 1912,

Bernstein [4] used the calculus of variations method to establish existence and

regularity results for the 2-dimensional real-valued Dirichlet problem. Serrin [25]

applied similar methods to extend these results for n-dimensions. We wish to

explore the minimization problem for u E BV(R, RN); in this case Du is a Radon

measure and we need give meaning to the variational integral (1-1).

For example, Giusti [17] considers a minimal surface area problem (Plateau's

Problem), where

F(p) :- + |l2. (1-2)

In this case,

SF(Du)dx= 1+Du 2dx

is the area of the surface of the graph of u.









Anzellotti and Giaquinta [1, 2], Hardt and Kinderlehrer [18] and Zhou [28]
study anti-planar shear of a thin plate (the plasticity problem). Here one seeks

solutions to the minimization problem (1-1), where




|p -3 if p > /3,

for some threshold 3 > 0. In this case, u : -- R is the displacement of the plate.
(C'!i Ihoolle and Lions [5] proposed a model to recover an image, u, from an
observed noisy image I = u + noise by

min F(Du) + (u I)2 dx,

for F(p) in (1-3). The diffusion from this minimization model is strictly perpendicular

to the gradient when |DuR > /, where edges are likely to be present, and isotropic
when |Du| <3 Thus the model preserves edges and eliminates noise.

Additionally, Chen, Levine and Rao [6] considered a function of q(x) growth,

for q(x) > 1, which is used as a model for image d vr..i-iir enhancement and
restoration. In this paper they proposed a new model for image restoration. The

proposed model incorporates the strengths of the various types of diffusion arising
from the minimization problem

min IDulp+ (U )2,
1n 2

for 1 < p < 2. In particular, they considered the minimization problem (1-1) with


SIplq(x) for p| < /,
F(x,p) := (1-4)
Ip1i q(x)- q() for p ,









where 3 > 0 is fixed and 1 < a < q(x) < 2. One may choose


q(x)= 1 +
1-+ k VG, (x)|

where G,(x) = exp ( |2/(4 2)) is the Gaussian filter and k,a > 0 are fixed.

In this case, the model utilizes a total variation approach when the gradient is

large (thus preserving edges) and L2 smoothing when the gradient is small (thus

removing noise). Furthermore, it employs anisotropic diffusion (1 < p < 2) in

regions which may be piecewise smooth or in which the difference between noise

and edges is difficult to distinguish.

Models based on minimization of a convex, linear growth functional of

measures have shown promising results in numerical implementation. The

development of PDE methods in image analysis is dependent upon answering

fundamental mathematical questions. In particular, the meaning of such a

functional of a measure and its first variation is not trivial to establish or

understand. While a definition for a convex, linear growth functional has been

given by Anzellotti and Giaquinta [1] and Giaquinta, Modica and Sou6ek [14], we

explore the motivation for this formulation and its relation to previous results in

convex analysis.

The functional

/ F(Du) (1-5)

is well defined on the Sobolev space W1'1(Q, RN). However, it has been shown that

the minimization problem for plasticity (F as in (1-3)), [1, 18, 28], and for the

image processing problem (F as in (1-4)) [6] has solutions u e BV(Q). This is

consistent with our intuition, as both models should allow for discontinuities (where

there is shear or edges, respectively). Since the solution is BV, its derivative, Du,

is a Radon measure. Thus we need to understand the meaning of (1-5) in such

a case. Indeed, the study of existence and partial regularity of solutions to these









problems depends on how one defines

/ F(x, u, Du),

for u E BV(Q). Anzellotti and Giaquinta [2] derive a formula for the special case

of the plasticity problem, with F as in (1-3). We take a different approach; we use

the double Fenchel transform to get a formula for the variational integral


I F(Du),

for a general convex function F with linear growth. Our approach has the

advantage of "naturalness" in the context of convex analysis. Additionally, we

establish some interesting properties of the double Fenchel transform.

We wish to establish some regularity results for solutions to functionals with

linear growth. In general, the minimum of a convex, linear growth functional

may have singularities, even if the integrand is smooth [12, 13, 14, 16]. In

applications, numerical results have shown the effectiveness of various convex,

linear growth functionals in feature-preserving image processing. Images restored

with these models are smoother in regions where the gradient is small and the

edges correspond to places where the gradient is large-a desirable property for

image denoising.

In particular, we shall focus on a problem that arises from anti-planar shear for

elastic-plastic materials,

inf F(Dv)- fv}, (1-6)
vEA [ I I
where v: t C R" -i R, the vertical displacement of material, belongs to an

appropriate set A and f is a given external force. F is defined in (1-3). Note

that F is a C1 convex function with linear growth. The minimization problem is

taken over the function space BV(Q), with either Dirichlet or Neumann boundary

conditions.






5


From numerical results in image processing (or deformation of a thin plate),

we see that the solution appears to be smooth when the gradient is small but not

when the gradient is large. Our goal is to verify this observation theoretically. That

is to -i-, suppose u is a minimizer of the functional (1-6). We wish to decompose

2 into sets E and P, where E is the elastic region, defined "-a.e. by


E ={x el: IVu(x)l < 1}


and P is the plastic region


P ={xRG : Vu(x)l > 1.


We expect u to be free of discontinuity and regular in some sense on E, but not

so on P. Furthermore, we note that it is not evident that we can remove a set B

of Lebesgue measure zero so that the set E \ B is open, nor is it clear that u has

no singular measure on E. However, if E is open, and u is free of singular part on

E, it follows from the standard theory that -Au = f on E. However, on P, the

minimizer u may have discontinuities, or behave like a Cantor function, even if it

is continuous on P (see Hardt and Kinderlehrer [18] for simple one-dimensional

examples).

We have followed the scheme set forth in an unpublished work by Tonegawa

[26]. While the general outline of the proof follows this work, we have provided

many missing details. In particular, we have provided a detailed proof for the

estimates of the Holder norm of a solution to an auxiliary problem, established the

first variation formula, and filled in details for the proofs of the main propositions.

Additionally, we have corrected a mistake in the handling of an estimate involving

the integral on the boundary by using the trace theorem for BV functions.














CHAPTER 2
CONVEX FUNCTIONS OF A MEASURE

2.1 Introduction.

We explore the meaning of jo F(m), for a bounded R'-valued measure. To

such ends, Giaquinta, Modica and Soucek [14] defined a function


F(x,po,p) := F (, o,

where x cE po > 0 and p E IR', and remarked that F is continuous on QxIR+xR",

convex in (po, p) and homogeneous of degree 1 in (po, p). They then proved that


lim F(x,po,p)
po-o0+

exists under appropriate conditions (given below). Let u E BV(Q) and choose

a positive Radon measure p so that the total variation, IDul, and the Lebesgue

measure, L', are absolutely continuous with respect to p. Denote the Radon-Nikodym

derivatives of L' and the vector-valued measure Du with respect to p by

dE' dDu
Sand d
d/j d/j'

respectively. They defined

I n I dn' dDu
F(x, Du) : F [x, dp.
I\ d dp )

By the homogeneity of F, it follows that this definition is independent of our choice

of p. The authors then applied a result of Reshetnyak [22] to this definition to

establish the lower semi-continuity of the integral.









Anzellotti and Giaquinta [1] applied this definition to the plasticity problem
with F as in (1-3) with 1. Here, the authors defined a function on R x R" by

F (P) t> 0,
F(t,p) :=
limt\o F () t t 0.

For any R"-valued measure m, consider the RI"+-valued measure a = (ao, m),

where 0 = is the Lebesgue measure in Q. The authors then defined

J/F(i) j f ( dao dm1)
0 F(m) : F[ d ) a (2-1)

where a is the total variation of a, and d and d- are the Radon-Nikodym
derivatives. In Lemma 1.1 [2], the authors used (2-1) to show that

I F(Du) F(Vu) dx + I D' (2-2)

for u E BV(Q). Here Du is decomposed into its absolutely continuous and singular
parts with respect to Lebesgue measure, i.e.,

Du Vu dx + Du.

To see that (2-2) follows from (2-1), we decompose Q with respect to the measures
Vu dx and D'u. That is to ;i, since the measures are mutually singular, there

exists a set A C Q on which Vu is not zero and D'u is identically zero. Thus on

A, Du = Vu dx and on the complement, Du = Du. Splitting the integral in (2-1)
into the sum of integrals over A and 2 \ A gives (2-2).
In a more general setting, Anzellotti and Giaquinta [3] provided a unified

approach to the partial regularity to solutions of the minimization problem (1-1)
for a general convex function F with growth m > 1; that is to i-, there are

positive constants a and 3 so that


a lplm < F(x,p) < 3(1 + |p|m).


(2-3)









In this case, the authors defined


J F(Du) = F(Vu) dx + F- (D |u (2-4)
0i (2 4)


where

F"(p) := lim F(tp).
t-oo t
Notice that this agrees with the definition taken above for the plasticity problem,

as F"(p) = pl in that case. In fact, one may use (2-1) and these techniques

to establish (2-4) in the general case. The definition (2-4) is used by Demengel,

Hardt, Kinderlehrer, Temam, Tonegawa and Zhou among others throughout
the literature in the study of existence and partial regularity of solutions to the

minimization problem with F as in (1-3) [8, 18, 19, 20, 21, 28].

While the definition (2-4) is sufficient for the study of functionals of this type,

there are still unsettled questions. In particular, what motivates this definition,

how does it relate to previous results and what is the relation to convex analysis?

The purpose of this dissertation is to take a very different route to the definition

of F(m) where F is a convex function on R" and m is a vector measure. Our

approach is to use Fenchel transforms to define fj F(m) in (2-4). Briefly, given a

convex function F on R", consider the convex functional on L1(Q, R") defined by

f I- F(f) dx.

We wish to extend this functional to the space of bounded vector-valued measures,

M. Additionally, we would like this extension to be lower semi-continuous on M in

the topology induced by the space of bounded, C(Q, R") functions, denoted by CB.

The Fenchel transform (or conjugate) of F on CB is defined by

F*() : sup (f F(f))dx.
f E L 10J









Since bounded vector-valued Radon measures are linear maps on CB, we can

naturally repeat this procedure on the space of bounded vector measures M, and

define

F**(m) : sup f d -F*).
6ECB J
Since L1(Q, IR) is a subspace of AM, F** is an extension of F. In this dissertation

we prove that the double Fenchel transform, F**(m) thus defined, is indeed given

by the formula


F**(m) = F(m') dx + F ( d ) d Im .

This result justifies and shows the "naturalness" of the definition of jf F(m) in

(2-4) in the context of convex analysis. One immediate consequence is the lower

semi-continuity of this functional. Additionally, this technique may be applied to

define convex functionals of objects more general than measures (e.g., certain types

of operators). Along the way we also find very interesting properties of F**. Just to

name one:

F**(m + n) = F**(m) + F**(n),

whenever the vector measures m and n are mutually singular.

2.2 Notation and Preliminaries.

Let Q be a bounded open subset of R". Let f e L1(2, R") and let F : R" -- R

be a continuous, non-negative, convex function with F(0) = 0, satisfying the linear

growth condition

lpl 7 < F(p) < pl + 7, (2-5)

for some a,3,7 > 0. Denote


MA {bounded -valued measures on Q}.









For any m e M, let ImI denote the total variation. We may decompose m into its

absolutely continuous and singular parts with respect to Lebesgue measure:

m = ma + M.


Denote

CB {( c C(Q, R) : is bounded}.

Endow L1 with the coarsest topology such that the mapping


L =(f) j f dx

is continuous for all Q E CB. Note that this topology separates points in L1

as ff Q = f g for all Q E CB if and only if f = g L'-a.e. With this

topology, L1(Q, R") becomes a locally convex, Hausdorff, topological vector space.

(LI(Q, RT), CB) is a dual pair for which we may define the Fenchel transform [7].

Suppose F satisfies the conditions above. If f E LI(Q, R"), the linear growth

condition on F gives

/ (a f )dx < F(f)dx < (/3 |f + 7) dx;

i.e., F E L1. The Fenchel transform, F*, of F is defined on CB by


F*(O) : sup f dx- F(f). (2-6)
f EL1,] I]I

Observe that for fixed f E L1, the mapping L6 is a continuous, affine map.

Hence F*(Q) is convex and lower semi-continuous on CB. Thus the desired lower

semi-continuity follows automatically from the Fenchel transform approach. See

van Tiel [27] and Ekeland and Temam [9] for more on the general theory of Fenchel

transforms.

We observe that

Proposition 2.1. Let Q E CB. If |1111 > 1, then F*(Q) = oo.









Proof. Suppose that 11011 > 1 and let f ,.,'-2 for k > 2. Since Q is
bounded, we have f c L1. Then

(f F(f)) dx= (,' F(.'-2)) d

> (. '- y) dx oo

as k -- o. D

For m E M, define the double Fenchel transform

F**(m) :- sup d F*( ). (2-7)
a CBJo

By Proposition 2.1, we may define

F**(m) sup f dm F*( ). (2-8)
a CBJo

If we endow CB with the uniform norm, then M C C The coarsest topology on

/M so that the mapping m -+ f 0 dm is continuous is called the weak topology on
M. By (2-7), F**(m) is lower semi-continuous in the weak topology and convex on

M.
For each fe L1, f dx cE M. It is easy to see
Proposition 2.2. Let f c L1(Q, R"). Then

F**(f dx) < F(f) dx.

Proof. By (2-8),

F**(f dx) sup f dx- F*(O),
eCB Jo

where F*() =supf L1 f(f F(f)) dx. For any f e L1 and every e CB, we
have

F*() > J(f F(f)) dx,
J/









so

I F(f)dx> J f- dx-F*().

Thus taking the supremum, we have


SF(f) dx> sup f dx- F*() =F**(f dx).
J6CB J
I16<1

D

Using the duality we started with, we can show that if F is convex and lower

semi-continuous, then

F**(f dx) F(f) dx,

for all f e L1.

Our goal is to derive an explicit formula for the double Fenchel transform F**.

This we hope will justify the definition given in (2-4).

2.3 Main Result.

We wish to show

Theorem 2.1. Let m E M1 be a vector measure. Decompose m into its it-... 'l. ,/

continuous part, m', and its .:,.ij.., part, m8, with respect to Lebesgue measure.

Then

F**(m) = F(m') dx + F ( ns ) d ,
Sh dl ldm )ll

where
1
F"(p) lim -F(tp).
t-oo t

Note that since F is convex and F(0) = 0, the limit in the definition of F"

exists.

Proposition 2.3. Let m, n E cM be ,in,,,il,' ..;/ .',jI denoted m I n. Then


F**(m + n) = F**(m) + F**(n).









We present the proof as a series of claims:

Claim (#1). F**(m) is lower semi-continuous in the sense that if mk -m n T..l.;,

in AM (i.e., j'f dmk -+ j' dm, for all 0 E CB), then F**(m) < liminf F**(mk).

Proof. For fixed G E CB, the map m v-4 fo 0 dm F* () is continuous. Since

F**(m) is defined to be the supremum of a family of such maps (for |1 < 1), we

conclude that F** is lower semi-continuous. E

Claim (#2). For rn, set K C 2, we have

F**(lK)
where we denote mlK = m K

Proof. Indeed, let (m, 0) denote the pairing Jo 0 dmn. We have


(m, 0) (nmlK + mlKc, = mlK, ( t) + (mlKc, ).

Therefore, for G e CB with 1l < 1, we have

(mlK, ) F( ) = m, ) F*( ) mlK-)
(2-9)
< (m, ) F*(0)+ ImlKt (Q),

as 101 < 1. The claim follows immediately from (2-8). D

Claim (#3). Let C E CB. For f E L1, denote

A A(f):= {x E : f 0}.

Then

F*() := sup f f-F(f)dx
fEL1 I
= sup (fl A(f)) F(flA(f))dx.
fEL1 f









Proof. Indeed, since F(O) = 0 and F(f) > 0, we see that for x E A(f) we


F(flA(f)) and for x ( A(f) we have F(flA(f))


F(0) = 0. Thus


4* (flA(I))


F(flA(f)) dx.


F*(0) sup 4- f
IEL


F(f)dx < sup J (f A(f))
f E L1 I I


F(flA(f)) dx.


On the other hand,


sup j. (flA)
fELi I


F(flA) dx <


sup f .
fEL1A


< sup fj
feL1 I


where L = {f e LI : spt(f) C A}.

Claim (#4). Let 1, 2 E CB. If 1I 112


0, then


A = A(f) = x E r : 1 f / 0}

Then for any x QE we have


F*(1) + F*(02).


and B B(f) := {x r : 2- f 0}.


1( 1 -f)(2 f)(x) < 1 11212 f2


Hence, for each x cE we cannot have both (01 f)(x) / 0 and (42 f)(x) / 0.


Therefore, A(f) n B(f) 0, for any f e L1.


have F(f)


F(f) > F(flA(f)). Therefore,


Thus


F(f) dx <


F(f) dx

F(f) dx


F*(01 + 02)


Proof. Let









Since A(f) and B(f) are disjoint for each f e L1, we have by claim 3 that


F*(01) + F*(02)


sup j (- (f) A) d
fEL1 f
+ sup f fB F(flB)dx

sup 1 (fiA) -F(fA)dx

+ j 2 (fiB) F(flB) dx



- (F(flA) + F(flB)) dx

sup j(0i + 2) (fIAUB) F(flAuB) dx
fEL1 I

F*(41 + 42),


as 1 (flAuB) 1 (flA + flB) 1 (flA) + 1" (flB) 1" (flA). Similarly,

2 (flAUB) = 2 (flB). Also note that since A n B = 0 and F(0) = 0, that
F(flA) + F(flB) F(flAuB). Therefore, F*(1i + 2) F*(1i) + F*(02) as
desired. D

Claim (#5). Let 0 E CB. Suppose that p E C(Q, R) such that 0 < p < 1. Then

F*(po) < F*(t).


Proof. We have


F*(4) -sup ".f- F(f) dx
fEL1 f
> sup / g F(g) dx
g pf J
fEL1
= sup j (pf) F(pf) dx
fEL1 I









> sup (pf) -f -F(f)dx
f6L1J

=F*(p).

For the second inequality, we note that for 0 < a < 1, the convexity of F gives

F(ax) < (1 a)F(0) + aF(x)

= 0 + aF(x)

< F(x).




Claim (#6). Suppose there exists a compact set K such that Iml (Kc) = 0. Let U

be an open set such that K C U. Then


F**(mr) sup d pdm- F*().
||<1 JQ
spt(6)CU

Proof. From (2-8) we have

F**(m) : sup dnm- F*( )

> sup /dm- F*(~).
|_<1 JQ
spt() CU

On the other hand, let p E C(Q, R) such that 0 < p < 1, p

on Uc. Then from claim 5, we have


I /dm F*(O)


1 on K and p = 0


< (O1K +1Kc) dm F*(p)

S 1K C) dm F(p (p



= p dm F* (p),
Jo.









as Im (K0) = 0. Therefore, we have


F**(m) sup f d F*()
I|11 Jn

< sup/ po dm F*(po)
|1<1 J

< sup / dm- F*(),
161<1 J
spt(0)CU

as {p : 0 c } C {c CB : spt( ) C U}. Thus the claim holds. Observe that

from the proof, we may also write


F**(m)= sup p F*(p).




Claim (#7). Let m, n e M.. Suppose there exist disjoint, compact sets K and L so

that ml (K) = 0 and In| (Lc) 0. Then


F**(m + n) = F**(m) + F**(n).


Proof. Let U and V be disjoint, open sets such that K C U and L C V. Let

pi, p2 C(Q, R) such that 0 < pi, P2 < 1 with pi 1 on K and pi = 0 on Uc, and

P2 1 on L and P2 = 0 on Vc. By claims 4, 5 and 6, we have

F**(m) + F**(n)- sup [ dm -F*( )
I16<1
spt(>)CU

+ sup [ dn- F*(O)
k16<1
spt(P)CV

sup pi dm F*(pil)
16i<1 L'O

+ sup p20 dn F*(p20)
|6|<1 LJQ J









sup pl dm + p20dn

- (F*(p) + F*(p))

sup i pidm + P2 dn
16|1 LJo JI
sup [ piO+p2 d(r + n)
161<1
sup [ d(m+ n) -
161<1 I
spt(6)CUUV
F**(m + n),


-(F*(pl + P20))]

- (F* ((Pl + P2)0))


Claim (#8). Suppose now that m,n E M4 are ,,,,inll,;/ :'ui,1.,,. Then

F**(m + n) F**(m) + F**(n).

Proof. For each j E N, we can choose compact di-i .1il sets Kj and Lj such
that Iml (Lj) = 0, In (Kj) = 0, Iml (K) 0, Inl (L) -+ 0 with mlKj m and

nall n, weakly in the sense of measure. By lower semi-continuity, we have

F**(m) < liminf F**(mlKY).

On the other hand, from claim 2 we see that

limF**(mlK,) < lim [F**(m) + Imn (Kj)]

=F**(m).

Whence, F**(m) = limF**(mlKj). Similarly, F**(n) limF**(nlL,). Furthermore,
since mlk, m and nlL, n, lower semi-continuity gives


F**(m + n) < liminfF**(mlKi + nlL).


as desired.









Conversely, by claim 2 we have

F**(mlK, + nL) = F** ((n + n)1KULj)

< F**(m + n) + Im + nl ((KL U Lj))

< F**(m + n) + |m| ((Kj U Lj)n) + In| ((Kj U Lj)).

Letting j oc, we have

lim F**(mlK + nrL) < F**(m + n).

Thus F**(m + n) lim F**(mlK, + niL,).

Therefore, by claim 7 we have

F**(m + n) lim F**(mlKj + nLj)

lim [F**(mlK,) + F** (ntL)]

lim F**(mltK) + lim F**(nL)

F**(m) + F**(n)

as desired. Thus Proposition 2.3 is proved. D

Let m be a vector measure. Decompose m into its absolutely continuous part,

m", and its singular part, m8, with respect to Lebesgue measure. Since m" I m8,

by Propositions 2.2 and 2.3 we have

F**(m) = F**(ma+ m")

SF**(ma) + F**(ms) (2-10)

< F(m a) dx + F**(ms).
Jo
Proposition 2.4. Let m E M. Then

F**(m) < F- ndn dm .
(/ d Iml
L E~|m|









Proof. Fix xo E Q and a > 0 and let a : -+ S'-1 be measurable. Consider

the measure au((xo)6o = m, where 6xo is the Dirac delta function. Then Iml = ao

and = m (o).

Let IB(x, r)| denote the Lebesgue measure of the ball of radius r centered at x.

Then we have
a^-B(xo,r)
al-(, (xo) dx aau(xo)b6o = m
IB(xo, r)
weakly in the sense of measures as r -- 0. Denote B = B(xo, r). From claim 1 of

Proposition 2.3 and Proposition 2.2, we have

F**(m) < liminfF** (au(xo) dx)

< liminf F a, (xo) t) dx
r-o J \ |B| }

= liminf F (ac (xo) t dx
r--0 JB \ \B\
r---+0|B

= liminf F a (Xo) )B
r--o |B
-F (aa(txo)),


by the definition of F". Moreover, for a > 0, we see that

F"(aj(xo)):= lim -F(ta7(xo))
t-oo t
lima
lim t-F(tau(xo))
ttoo ta
SaF" (u(xo))

a I F{ (a(x)) d6o
Jo
SF-(,-(x))d m| .


Thus we have

F**(m) < F" (,(x)) d m (211)
Jo









Now let xl, x2,... xk be a finite set of distinct points in Q and ca, a2,.. ., ca >

0. For each i, denote a(xi) = ui E S"-1. Denote the measure airi6x = mi. We will

refer to a measure of the form
k k
m ai(xi(Xi)6Jx Zmi
i= 1 i= 1

as a simple measure. Observe that mi I mj for i / j. Thus by Proposition 2.3 and

(2-11), we have

k
F**(m) ZF**(m,)
i= 1
k

i 1

F/ -( dm


For the last equality, we need the fact that d Imi = ai d6x, is a weighted point

mass, for distinct xi. In such a case, we have the desired additivity. Hence for a

simple measure m, we have shown


F**(m) < F- d dml d m .
d d d m.

We now extend this to a general measure:

Claim. For i,, m E .Ad, there is a sequence {mi} of simple measures such that

mi m ,a'..',;l and

F ( dm ) d ir foo-rF dmi ) .
/ F" ~ dr d|m|\= lim F" d |m|.
d |m|- d | dmi| 1

As a limit of continuous functions,


F"(p) lim F(tp)
t-oo t

is measurable. Let a : Q -- S'-1 be a measurable function such that ad Iml

dm. Then F"(a) is measurable as well. Since F is convex and continuous and









!F(tp) is increasing in t, convex and continuous, it follows that F"(p) is lower
semi-continuous and convex. Since F has linear growth, we have

0 < F"(p)
By a standard result on convex functions, we conclude that F" is indeed

continuous. So by Dini's Theorem, -F(tp) converges uniformly on compact sets.

Let 01, 2,.... be a sequence in the unit ball of the Banach space C(Q, R")
with the supremum. Then for any p and any sequence p, E AM, we define p~ p

weakly if and only if

lim (0j, P.) = (0j, P),
n-*oo
for all j.

By Lusin's Theorem, for each k we may choose a compact set Ck such that
Iml (Ck) < ), a and F"(a) are continuous on Ck, and Ck-1 c Ck. Since a, F"(a)

and Oi (1 < i < k) are continuous and thus uniformly continuous on Ck, we may
choose 6k such that

\7(x) 7(y)\ + \F({a (x)) F- (a(y)) \ + \(x). (x) i(y) U, (y)\ < -
i= 1
whenever x, y e Ck with Ix y| < 6k. Split Ck into disjoint subsets Ai,k (1 < i < ik)
such that diam(Ai,k) < 6k. Pick xi,k e Ai,k. Then the sequence of simple measures


nk i u7(X,k) mn (Ai,k)6,
i 1
satisfy the claim. That is to i-, we have

d/r/ndmk 2 ||F"|| ||m||
LF"(jT) d) m I d d~mk| k

where |IF"I = supply IF-(p)l. Thus the claim holds and the proposition is

proved.








From (2-10) and the previous proposition, we have

F**(mn) < F(m') dx + jF ( dm ) d' (2-12)

Proposition 2.5. Let f E L1. F** is the 1',/ -/. convex functional on M such that

F**(f dx) < F(f)dx.

Proof. We will show that F** is the largest convex functional on M in the
sense that if G is also a lower semi-continuous and convex functional on M, and
G(f dx) < fQ F(f) dx, for all f c L1, then G(m) < F**(m) for all m e M.
Indeed, let m E M and suppose that G is lower semi-continuous and convex
on M such that G(f dx) < f' F(f) dx. Let A < G(m). Then by the Hahn-Banach
Theorem there exists 0 e CB and a number K > 0 such that for all n AM,

G(n) > fdnt-K, and A< I dm K.

In particular, for any f E L1, we see that

G(f dx) > f fdx -K.

However, G(f dx) < f f F(f) dx by assumption. Therefore,


I F(f) dx > f dx K.

Thus, for all f c L1, we have

S. f dx- I F(f) dx < K.









Taking the supremum over f E L1, we conclude that F*(0) < K. Thus for E CB

chosen above and all n c M we have

F**(n)- sup j -dn F*()
||5i Jf
> / dn F*(O)

> j dn K.

In particular,

F**(n) > dm n > A.

Since A < G(m) was chosen arbitrarily, it follows that

G(m) < F**(m).

In fact, if we suppose that G(m) > F**(m), we may choose A such that G(m) >

A > F**(m), a contradiction.

D

Finally, it remains to show that

Proposition 2.6. Suppose that F is convex and m E M. Then

F**(m) = F(m') dx + JF ( ds ) d m (2-13)

Proof. By Proposition 2.5, we need only show that the right hand side is

convex. Since F is convex, we know that F" is convex as well. For the convenience

of the reader, we show the convexity of the map

m F- F(m' dxd + F ( dm d (2-14)

The absolutely continuous (singular) part of a sum of measures is the sum of

their absolutely continuous (singular) parts. Thus we need only show the convexity









M f F- ( d1 )dm j \ ,,
d (m d

To this end, let m, n E M/ be measures and let 0 < t < 1; denote s
brevity, denote


d(tm + sn)
d Itm + sn
d Iml
td iml + sd n


(2-15)

1 t. For


d itm + sn|
td ml + sd n
d ln
td ml + sd In


Then


F (( d(tm + sn)) dtn+sn
d Idtm + snjl

00 d itmn + sn)
( F (d (Tm + s n)
F dIT- + sB (td || + sd I|n)




(as F"(an) -= Fc(n) for a > 0)

S~F~d tdm + sdn


S td(td ml | + sdd n
+sFc ( |dn (td|Im+sdn)
G d |m| + sd In|

(by the convexity of F')

t F- C (td |m +sd |n|)
j ((( d Iml
+s F- ( D) (td m +sd|n|)
In d (ld In





26


=t I F- dd ) C(td m + sd n|)


( d/ml d
+s (F D(tdm| +sd|n|)

St JFO (dd) dm| + s jF" (dt)d |n,

proving the map (2-15) is convex. O

The equality (2-13) is precisely the result we have set out to establish, and
thus Theorem (2.1) is proved.














CHAPTER 3
A PARTIAL REGULARITY RESULT FOR PLASTICITY
3.1 Introduction

Below, we establish a partial regularity result for the plasticity problem (1-6);
for simplicity, we will take 1. Let Q C R" be a bounded domain with Lipschitz
boundary, and functions f E L'"() and p E L'l(0) be given. For u E BV(Q),
we decompose its gradient measure Du into its absolutely continuous and singular
parts with respect to Lebesgue measure: Du = Vu dx + D'u. From our previous
work, we may define


I F(Du) F(Vu) dx + D'u ,

with the boundary condition o appropriately satisfied. The existence of a
minimizer was discussed by Hardt and Kinderlehrer [18] for the Dirichlet problem.
There exists a minimizer if IlfllL < CD(Q)-1, where CD is the smallest constant
such that

S1|L1(0) < CD() ID(| ,

for all ( E BV(Q) with (Qan = 0. The condition (the so called safe-load condition)
guarantees that the functional is bounded below. Thus a minimizer exists by the
lower semi-continuity of the functional. Similarly, for the Neumann problem, the
condition IlfllL| < CN(Q)-1 guarantees the existence of a solution, where CN is the
smallest constant such that


|4L1(0_) < CNw() ID(|,
Jo









for all ( c BV(Q) with f ( dx = 0. Zhou [28] studied the parabolic problem

associated with the above functional. Hardt, Tonegawa and Zhou [20, 21] study the

related gradient flow, where F is convex and has linear growth.

Below, we show the following regularity result at a point where the solution is

close to a linear function with slope strictly smaller than one:

Theorem 3.1. Let u be a minimizer of the functional (1-6) with either Dirichlet

or Neumann boundary condition. For i,.;i given 0 < p < 1, there exist positive

constants co and Ko, which depend only on n and p such that, if

t/- IDu -l IBr J., a)

holds for some B,(a) CC Q and for some 1 E R", with


r f IIll -< Ko and Il\ < 2/,

then

IDul (B,/2(a)) 0, Vul < 1 p on B,/2(a),

and u solves

-Au = f on B/2 (a).

In particular, u C W2 (B,/2 (a)) for ,,i., p < oo.

Note that by the Sobolev embedding theorem, we have u c C1'"(Br/2(a)) for

any 0 < a < 1. It follows by a standard result in measure theory that

Theorem 3.2. Let u be a minimizer of (1-6) with Dirichlet or Neumann bo;,,i; i..

condition. If "({IVu| < 1}) > 0, then there exists a no,. m1iil; open elastic region

E on which u is in C1',, |Vu| < 1 and u .,I;-7.

-Au = f on E.


Moreover, IVu(x) > 1 for ~"-a.e. x C Q \ E.









We note that the previous two results depend only on the minimizer u and

not on the initial or boundary conditions. However, it may happen that the

"-measure of the set {|Vu| < 1} is zero for both the Dirichlet and Neumann

problems. The following conditions on the boundary value p and the force term

f, which are physically reasonable, assure that the there exists a nonempty, open

elastic region.

Theorem 3.3. There exists a constant co = c(Q) such that, if


II|hL1(aQ) < Co and I\f |L < Co,

then i.:;' minimizer u for the Dirichlet problem has a no., ml,;ii open elastic region

with the properties stated in Theorem 3.2.

For the Neumann problem, a restriction of the size of the force, f, guarantees

the existence of a nonempty elastic region.

Theorem 3.4. Let cN(Q) be the smallest constant such that


(L\(Q) < CN/ D(\,

for all ( E BV(Q) with jf (dx = 0. If |If |ll < cN/2, then ,:,;1 minimizer for the

Neumann problem has a no. ;,1i/;, open elastic region with the properties stated in

Theorem 3.2.

The key to the proof of Theorem 3.1 is the decay estimate given in Proposition

3.3 below. We show that the functional 4, defined below (3-7), which is an average

of the gradient measure, decays for small balls with a linear correction. We achieve

the decay by approximating a minimizer to (1-6) by a Lipschitz function and

establishing an L" estimate for their difference. The linear correction arises by

approximating u with the solution of an appropriate PDE.









3.2 Decay Estimate

Below we fix p > 0 and denote constants depending only on n and p by cs.
Our first lemma gives a lower bound for the functional (1-6) for some function u of

bounded variation in terms a function with gradient strictly smaller than one.
Lemma 3.1. Let u e BV(B,(a)) and h E C'(B,(a)) n W1,2(B,(a)) with

supB,(a) Vh < 1 p, then


F(Du) F(Vh) dx> V(u- h)-Vhdx+ -
rJ Br JBr 2
+1 V(u -h) 2 dx+ +
S 2 a Bn{|vul
Proof. For the part where I|Vul > 1, we have


IBrn{IVU l} Vul dx

r DSu + IBr DS'u- Vh
3, JB,


F(Vu) F(Vh)-V(u -h) Vh
1 1
= Vu| + IVh2 Vu Vh
2 2
1
> (2 Vu- l-IVhl)(1- IVhl)
2
> 1VuI 2 U2
2 2

For the part where |Vu| < 1, we have

F(Vu) F(Vh)-V(u h) Vh
1 1
I Vu2 Vh12 (u h) Vh
2 2
1
SIV(u- h)12
For the singular part we have
For the singular part, we have


B, D u >r / D)uVh + J, D1 ( -

> f DSu Vh + p Du ,
B BR


|Vhl)


l









since IVhI < 1 p. Hence, by combining the three inequalities above the desired

estimate is deduced. D

Let f e L'(Q) be given, and fix B2r(a) CC Q. Also let v E C,'(B2r(a)) be

given. Below, we use the smoothing of Lipschitz functions adopted from Schoen

and Simon [24]. For 6 > 0 and 3 > 0, which we will choose later, assume that there

exists 1 E RW such that


sup IVv I < 826 and I|| < 1 2p.
B2,(a)

We denote v(x) = v(x) 1 x. Let b E Co'(R") be the usual radially symmetric

mollifier with compact support in Bi(0) and with sup(| | + |V |) < cl. Denote the

scaled mollifier by '' ,(x) = a- (x/a) and let


Va = ra IV


and Va = bra v


be the usual convolution. We then have the following estimates [24].

Lemma 3.2. Using the notation above, we have


(i) sup |Vv3
B,(a)

(ii) sup vp -
B,(a)

(iii) r6 sup x -
B,(a)


< c2r


-I | sup |Vv| < 026,
B,(a)

v = sup VU vI < r0 sup |Vl| < r/ +2,
B,(a) B,(a)

- yl-6Vv(x) V3g(y)

6 sup V -V / sup y 1 (x/(rs)) (y/(r3))|
B,(a) x,yEB (a)


< c3o26 -6 < C3 36.









Proof. Let v E C(' (B2 (a)) and use the above notation. To verify the first
estimate, for x E Br(a), we have


|Vv(x) l=V("' :,v )(x) l '/ :* Vv(x) l

j (x y)[Vv(y) l]dy
J -yl |(, ^* Vv) (X) -V ( V)( (x)

V=13(x)I

Hence suPB,(a) |Vv l = sup,(a) IVv3I. From (*), it follows that

|VvO(x)| < sup IVv f rP(X- y) dy < sup IVv /l < 32
B (a) Jix-y <
as desired.
For the second inequality, we observe that

vp(x) v(x) ( : v)(x) v(x)

( y) [v(y) 1 x] dy (v(x) 1 x)
Svx-y v V3(x) -V(x).


6


Thus suPB,(a) Iv3P I = SUPB,(a) IV3 V0.
Furthermore, by the mean value theorem, there is a ( E Br,(x) so that

(X) v(x) :(y -x) [v(y) v(x)] dy

JIx-y\_ by<. 1 ho(Y -x) \Vv(o\ y -x dy

< (r3) sup |Vv\ < 26r =- r031+26,
B (a)

by the hypothesis on v. Taking the supremum, the desired inequality follows.








Finally, for the last estimate, let x 1 y E Br(a). Then we have

|Vv3(x) V3(y)|I I|Vv(x) V3(y)|
x y1| x y y
x- y 6 (,3 r(x- z) (- r3(y z)) (Vi(z)) dz
JRn
< sup VI x y- (x z) :(y z) dz
B, (a) J
= sup |VV| |x y|-6 (ro) ( z) dz


= sup IV7 (1rR)-" R dz
B,(a) JRn |(x Z) (y Z)|I

(letting x' = x z and y' = y z,)


< sup IVvl (rp)- 2(r/)" sup )
B,(o) xX 7Y' I' y'l
= c sup |Vv (r/)-6 sup
B, (a) / 7Y' x' y'l
< C3 26 r-6 -6 c3r- 6.

The desired estimate follows immediately. D

From the usual theory of Poisson equations (see Gilbarg and Trudinger [15],
for example), for any r [j, r], there exists a unique solution w C W1'2(B (a)) n
C1'(B,(a)) for
-Aw f on B,(a), (3
(3-1)
w = v3 on aBf(a).
Moreover, we have the following estimates.
Proposition 3.1. Fix r E [r, r] and let w be a solution to (3-1). Then

|7Vw(x) Vw(y)|
(i) r sup Vw) < c4 IfL I + 1 61,
B,(o) X Yl1
i s |IVw(x) Vw(y)| < c -1 z + I/
(ii) sup WX) W(y < C5 |v3 dS + r1/2
BF/2(a) X y\/ \r+ jaBF(a)









Proof. Let w be a solution to (3-1) and write w

w1, w2 W1,2((a()) n C',( Ba)) so that


wi + w2, where we have


f on B,(a),

on OB,(a);


Aw2
and A
W2 =


= 0 on B,(a),

vp on OBf(a).


For wl, we may use Green's function to write


1
n-2f(z) dz.
- z|


Differentiating, we get the estimate


Vwi(x)- Vwi(y) I
x- y




For w2, we let w = w2 v3 and consider the resulting problem


SAw

w=0


-Av, on B,

on OB,.


We have (rescaling for B,) that


r [Dw]6,B, < c || '||(B,) + ||Vt ||() + r6[VV]36,B,

where [ ]as,B denotes the H1lder semi-norm on the ball of radius r with exponent 6

[15, Theorem 8.33].

It remains to estimate the right hand side only in terms of the H1lder norm.

We have [15, Theorem 8.16],


II' L-m(B,) < sup C+c C I|| i l| (,) c | C Vt IIL ,).
8Br

Combining these last two inequalities, we have


r [D]a,s,B < c [|Vt I||L() + r6[VVa,,]


Awl -

w= -0


W1 (X) B (a) IX
I X









Replace vp by vp = v3 1 (x a), where 1 = Vvg(y), for some fixed y c B,(a).


Note that AVp


Av3 and [VVP]S,B,


[Vv316,B,. We claim that


II 3 1 L|<, ( cr[V, ],,.

In fact, for y E B, fixed above and any x e B,, we have Ix y| < 2r. Therefore,

Ix y6 < (2r)6. Thus


V ) V ) (2r) Vv(x)
I|vp(x) wV (y)| <


ix-y


- V3(y)l
I i


Whence


IVv3 1L-(B,) <


cr sup
x,yEB,
X7'Y


IVvP(x))
Ix


y 7 ([V ]V (
v\


Thus we have

r6[Dw],6,, < c [|| V -/ L (,) + r6 [Vv]36,] < cr 6[Vv]36,, < c6,

where the last inequality follows from Lemma 3.2. It follows immediately that

[Dw216,B, < cr-6 6.

For w = wi + w2 we assemble the above pieces to see that


IVw(x)
[Vw]s,B, sup
Bf(a) X


- w(y)l
-y1


Wsup l(x)- Vwi(y)
B< sup 1,
B,(a) |x- | y


IVw2(x)- 72(Y)
|x- y1|



as desired.

For (ii), we note that for 6 = 1/2, the first part gives


[VWi]1/2,B/2 < [VWi]1/2,B, < C lflL| (B) r1/2









For w2, we use Green's representation


w2(x) = K(x, y)v3(y) dSy = v 2 X12 3(y) dS,,
JB, (0) JB9 (0) n rI X iI

where w is the size of the n-dimensional unit ball. For simplicity, we drop the

subscript and write w2 = w. Taking the derivative, we have

i'. (x) L= B Kx(x,y)v3(y) dSy.
J(JB,

Therefore, for x,x e B,/2(0) the mean value theorem gives

,,, (x)- ,,,,( ) r v ( ),
2- 0 ) Ix X I v2(y) dS,,
X X S B1/2 X X1/2

for some B E Br/2(0). Estimating IVKx(,(,y) we have

12)- ( < VK(xj,y)| Ix |1/2 v3(y) dSy
\X '- 1\ Br
|1x Xx |)
S--1 r2 jBr y)\ dSy

< I 1v1 dSy.
I QBr

Thus for x, y Bf/2(a) it follows that

|Vw(x) Vw(y)| |< Vwl(x) Vwl(y) IVW2(x)- Vw2( /)
S11/2 X 11/2 + 11/2
x- y y x y

< c (r1/2 If / L + t B, |v1 dS

as desired.

D

Remark. It follows from Proposition 3.1 that

sup |Vw l < c (f + r I|fl|l) .
Br(a)








Indeed, for x E B,(a) and y E aBf(a) we have


IVw(x) I| < IVw(x) Vv3(y)| + IVvp(y) |I
< (2r)6 IVw(x) Vw(y) I| ()
x,yEB((a) Ix y v

< c (6 + r I||l|L ),


where we have used Lemma 3.2 to estimate |Vv'(y) 1 < 326.
Lemma 3.3. Suppose that v c C,'(B2,(a)) and that there exists 1 C R" such that
Il1 < 1 2p and supB2 (a) |Vv 11 < 326. Let v3, r and w be as described above.
Then there exists constants c6 and c7 such that, if/36 < c6 and r IlfllL < c7, then
the following .,:,. ,;,,';,: ,i holds:


u) F(Vw) dx > (u v) OdS+ (u w)f dx

+j + 2 n I vul 1}ul dx + IV(
IJ JByn{IVul>l} F Bn{Ivul

u- w)1 dx.


Proof. From the previous remark, it follows that

suupVw < s Vw /| + ||I < c(036 + r |f |I||) + 1 2p.
BF(a) Bf(a)
C' ....!- the constants c6 and c7 so that, if 36 _< C6 and r |I fLL- < c7, then

c (36 + r ||f ||L) < P.


Therefore,


sup |Vw| < 1 p.
B,(a)


/ F(D
JBy








We may now apply Lemma 3.1 with w in place of h to get


/ F(Du)- j F(Vw) dx

> J V(u- w) -Vwdx+ J D Vw + p ID

+ -2 Vu dx + t1 IV(u w)12 dx.
2 gB n{|Iuai>} 2 JBn{ Vul To get the desired inequality, we integrate by parts as follows:


I V(u- w) Vw + / D'su Vw
= D(u w) Vw


(U v) OdS
BaB On


S(u- w)f dx,


as DSw = 0. Substituting above, the lemma is proved. D

A function u E BV(Q) is said to be a local solution in Q provided that

F(Du) fudx < F(D(u+ ))- f(u + () dx, (3-2)

for any ( E BVo(2). Observe that a minimizer for the Dirichlet or the Neumann
problem is a local solution. We note that the following argument requires only that
u be a local solution.
Next, we establish a first variation formula by computing the Euler-Lagrange
equation for the functional (1-6).
First Variation. Let u E BV(Q) be a local solution and Q = Qa U Qs be the
decomposition of Q into sets Qa, where Du = Vu, and Q,, where Du = Du. For
I,,' ( E BVo(Q) -,'i/'fying DS( < ID we have the first variation formula


I f(dx,
Jo


I a J V(dx + I a | Du








where a = o(u) is the stress tensor /, fI;,. by


F,(Vu) on fa,
o- (>) D=
DMu
Son,

and is the Radon-Nikodym derivative of D'( with respect to |Du|. For -iil. I/,
we i,,;'1 denote the first variation as

I a- D( f ( dx.

Proof. If u E BV(Q) is a local solution, then for t > 0 we have


I F(Vu) dx + I D fu dx
< F(V(u + t())dx + D'( u+to f (u + t) dx.

Moving everything to one side of the inequality and dividing by t > 0 yields

0< JF(Vu+tV F(Vu)dx + [ ID(D +t()- Dt ] f(dx.

Since Dc' < IDul taking the limit as t O+ we get


0 = Fp(Vu). V( dx + I D


Sf( dx.


Since,
SIDD D'u Ds ( D

the first variation formula follows immediately.

Note that by Dirichlet's principle (see Evans [10]), it now follows that u
satisfies -Au = f, when |Vul < 1 and u is free of singular part.








Lemma 3.4. Let w be a solution to (3-1) and suppose that u is a local solution.
For F [C r], we have

j F(Du) F(Vw) f (u w) dx < f u v| dS.
JB JB J Bf JBBF
Proof. Recall that w E W1'2(B,) n C''(Bf). Let

w in B,

u in Q \ Bf.

Note that w has the same boundary data as u and that from the usual theory of
functions of bounded variation (see Evans and Gariepy [11], for example) we have
w E BV(Q). Thus from the definition of local solution, we see that

SF(Du) I fudx< IF(Dw) fw dx

/ F(Vw) + F(Du) + F(Dw) (3-3)
JBr JO\BF J8Br

-B f w dx- B fu dx,

since Dw = Vw. Since fJB F(Du) + faB IDSu >fB J F(Du), (3-3) may be reduced
to

/ F(Du)- F(Vw)- f (u w) dx < F(Dw).
BF Bf BF QBf
It remains to check that


f F(Dw) < f u- vL3 dS.

By the trace and extension theorems [11], we have that

Dw = (Tw- Tu)rld "-1 on 9B,,

where rT is the outward pointing normal vector to 9By, and Tw and Tu are the
traces of w and u, respectively. By the definition of F(p), we have F(p) < Ipl for all








p, so that


i F(Dw) JaBy JaB, JaB,
Since Tw v3, we may (by slightly abusing the notation) write


B I (Tu Tw) d-( 1
The desired inequality follows.


I/ u vf| dS.
J8B,


Lemma 3.5. Let r E [, r] and suppose that v E C'o,(B2 (a)) is as in Lemma 3.2.
If u c BV(Q) is a local solution and w is a solution to (3-1), then


/ F|D u + I B\ {Ivl_>1}
J B J Brn{\vu\l1}


|Vu| dx + /IB vul JBen{|Vu|

|V(u w)2 dx

< c j vl dS + cg91f26.
JBf


Proof. By Lemmas 3.4 and 3.3, we have


IB |u-v3 dS+ f(u -w)dx> F(Du) -
OBw

B1 2 wB 2
+ V( w) 12d.
2 JBFn{\vu\

j F(Vw)

w) dx

Vu| dx


Therefore


SI 2+ Vu d 2 \n u l IB> n{ vul
(Uw wF {Z
< u- V + D u- v- | dS + dS
aBs, OB aJS s









< c (j u v dS + Iv V3 dS)
\JB J8By /
< c |u v dS + c'r"-1 (r3l+26)

as the remark following Proposition 3.1 gives | < /1l + c / + r Ilfll) < C

and we have Iv v3| < r/l1+2 by Lemma 3.2. The desired inequality follows
immediately.



Below we show that if u is a local solution and v is a Lipschitz function with
small gradient that coincide except for a set of small measure, then we can estimate

IIU vIIL. The result is a modification of Hardt and Kinderlehrer ([19, Theorem
2.2]), which we present here to illustrate the important changes.
Proposition 3.2. Suppose that u E BV(Q) is a local solution in Q and that

B2r(a) CC Q with r If ILoo < C7. Let v E Co0'(B2r(a)) such that supB2,,() IVv <
1- p and

({u / v} n B,(a)) < IBp for all r < p< 2r.

Then there exits positive constants clo and ll so that if

({u v} n B2(a)) C10rn

then

IIu vIIL| < ell(' ({u / v} n B2(a)) )l/n

Proof. With out loss of generality, suppose that a = 0. We then center all
balls in the discussion below at 0 and note that x/Ixl denotes the outward pointing
unit normal vector. Let u be a local solution and let 0: R -+ R be a bounded,

ii i., -ii- piece-wise differentiable function with O'(t) < 1 for all t, which we will








define later. Suppose that 0 < p < h < 2r and define

1 in B,,

x) < (h- p)-(h Ixl) in Bh \ Bp,

0 in Q \ Bh.


Applying the variation formula fj ca D(
product rule gives


j f( dx with (


rO(u v), the


/ D[O(u- v)] -(h- p)-1 e-O(u v)dx
Bh JBh\B xi


-B / h O(u-v)f dx. (3-4)

Observe that by convexity,

F(Dv) = F(Vv) + |D'v < Fp(Vv) Vv + D'v = a Dv,

for any v e BV with D"v < ID|u. Also, by the linear growth constraint (2-5),
there exists c so that pIl < c(F(p) + 1). Recall that Dvl = |Vv| < 1 p. We have
the following estimate:


SI D[0(u v)]
Bh


< / ]O'(u v) IDu Dv\
h
< b rTO'(u- v) Du + L ry0'(u- v) Dvl
Bh Bh
Bh Bh
< L T- D[O(u v)] + Lf \0'(u- v)a Dv\
Bh Bh
+c -rlO'(u v)
JBh

JBh h








Substituting (3-4) into the above inequality, we see that


/ T\D[O(u-v)]|

< (h- p)- B, 0(u v)| +f \(0 -u v) f + c O'(u v)
JBh\Bp h h
< (h p)-' | ( v)I + Ifll || |O(u v)| +c |spt(,O(u v))1,
JBh\Bp Bh
(3-5)


as spt(Tl) = Bh, and 0'(t) < 1.
Let 0 < k < s < oo and define


0

(t) = < t
s~> (


Define


A(k,h) : Bh

and note that Ispt(rO(u v)) = IA(k, h)


SID[0(u- v)] <
Bp


for t < k,

for k < t < s,

for t > s.


n {|u v| > k}

Since

JB ID[O(u- v)]1,
Bh


we have from (3-5) that

/ D[O(u -v)]1 <(fI + (h -p)-1) 10(u -v) +cA(k, h).

Note that from our definition of A(k, h) and the conditions on u and v that

IA(0,p) I ~({u v} nB)< l Bp ,


for all r < p < 2r.









From the definitions of 0 and A(k, h), and the isoperimetric inequality (see

Evans and Gariepy [11, Theorem 5.6.1, part iii]) we have

n-
(s k) A(s,p) I _< (10(u- v)| dx)

< cf ID[O(u- v)]
JBp

< c (| f + (h- p)-1) f O(u- v)| +c' A(k, h)
J Bh
< c [(h p)-(s k) A(k, h)l + IA(k, h)].

Whence
n-1
IA(s,p)l < c [(h p)-1 + (s k)-1] A(k, h) (3-6)

for r < p < h < 2r and 0 < k < s. To deduce the desired result we use Lemma 2.1

from Hardt and Kinderlehrer [19], which we state here.

Lemma 3.6. Suppose

{A(k, h) : r < h< 2r, k > 0}

is ,Ii, collection of subsets of B2r that ri, fy (3-6) and that 7 is a positive number

with < < | B1 and 71/ < cl6. If A(0,2r)| < r', then A(d, r)| 0, for

d C16 IA(0, 2r) |1/.

By the lemma, if

IA(0, 2r)| -= ({u / v} n B2r) < cF,

then IA(d, r)| = 0, where


d c16 |A(0, 2r) 1|/ c16 ( u({ u v} n B2r.))1/n









But A(d,r) = {u vI > d} n B,, so IA(d,r)| = 0 implies that lu v| < d on B,.
That is to ;i- that


IIU vl|(B,) < d= el (J({u / v} n B2r))1I

as desired. D

Remark. To apply Lemma 3.6 to the proposition, we take co = 7 so the condition

({u / v} n B2,) I A(0, 2r) I <

implies that


IIU V\ L\ < C11 (({u / v} n B2r))1in C11 |A(0, 2r)| d,

if we take cll = c16
In the following proposition, we denote

4(r,1, x) := V \Vu( dx
I l (x)n(3 7)

+ |IVu 2 dx + I D' ,
JBr,(x)n{vul<1} JB,(x)
for r > 0, 1 C R" and x e Q.
We show that one can find a Lipschitz function v that approximates a local
solution u. C' i.... -i ,g a ;ood slice" of ball and assuming that 1 is small, we show

that 4 decays for a smaller ball, with the addition of a small linear correction.
For the proof of the standard Lipschitz approximation (steps 1 and 2, below), we

reproduce the proof of Theorem 2 in section 6.6.2 [11] with some modification to
suit our needs.
Proposition 3.3. Suppose that u is a local solution and that B4,(a) CC Q. Let

l1 C RW be a vector such that I111 < 1 2/. Then there exist positive constants u', e,









K, c8l, C19 and C2o, depending only on n and p, such that if

t (4r, i1, a) < e,


for r with r( f \\L- + 1) < K, then there exists 12 CE I for which


t + C1sr211 f 11 2^
)i(Lwr,1 2, a) < 1 1(4r,lI, a) + CL-r2 f

Moreover,

11 /2 < c191(4r, 11,a) 12 + C2or IIfl L

Remark. The proof is long and technical; our approach will be to break it into

several steps:

1. Define a set RX where (p, li, x) is small and estimate (B2r(a) \ R).

2. Let g(x) = u(x) 11 x and show that g is Lipschitz on RA. Using a standard

extension theorem, we establish the existence of a Lipschitz function v on B2r

such that v = u ~-a.e. on RX.

3. Use step 1 to estimate the size of {u / v}.

4. Estimate fo lu v| dS in terms of 4(4r,11, a) and apply Lemma 3.5.

5. Assemble the above pieces to get the desired estimate on (wUr, 12, a).

6. Note that in the process of the proof, we have what we need to obtain the

desired estimate for I|I 12 .

Proof. Step 1. For 0 < A < 1, which we will fix later, define

R = {x E B2r(a) : (p, 11, x) < A for all 0 < p < 2r}.

By Vitali's covering theorem, there exist disjoint balls {B,,(xi)}), so that
00
B2R(a) \ RX C U 5rji(i) and 4 (ri, l, x,) > A.
i=1









Thus, we can estimate

LT' (B2 (a) \ 'R)< Bj (Xi) = 5" r, (xi)
i 1 i= 1
5n "C
< lB (r)(x |l (r i,,lx)
i= 1
5"
A
< I|B4,(4r, 1, a).

Step 2. Let g(x) = u(x) 1 x. Below, we denote the average



Usg H e g(y) dy.
sing lders and Poincares inequalities, for any (x

Using Holder's and Poincare's inequalities, for any x E RA and 0 < p < 2r, we have


cp
\Bp A Lrn(BP)

- n-1 IDgl

[ I Vu -lI dx

Sn-1 Bp()Vu
IVU
Pn-1 JBp(x)n{\vu_>l}


+ JB()


S+ |11 dx +
JBp(x)


+ |Vu lllL2(Bp(x)n{|Vu|<1}) ||1L2(Bp(x)nj{Vul<1})]

p- [ 1Bp| (p, lx)+ |Bp1/2 (p l (pl 1))/2


where we used the fact that all the terms of (p, li, x) are positive and 0 < A < 1.

We note (for use later in step 5) that we have, in the process of establishing the

above inequality, shown


C I Dgl


(3 8)


c IDu I < C22 <(p, 11, )1/2
P" Jn


ID'u


IDul









It now follows that


(9)x,p/2 g,p/2 9g(Y) gx,p/2k dy
S Bp/ 2k+ B(X) +l (X)
2"2 r
< g(y) (g)x,p/2k dy
B p /2k() JB 2 (x)

< 2"22 1/2 (P\ C 23A 1/2
< ~k 2 2k

Since g E L1(B2,(a)), we conclude from the Lebesgue point theorem, that g(x)

limpo(g)x,,, for L"-a.e. x e B2,(a). Whence, for a.e. x E RA, we have
00
|g(x) (g)x,p < (9),,p/2+l (9g),p/2k < C23pA1/2.
k-0

For x,y E RA with Ix- yl < 2r, set p= x yl. Then for a.e. x,y RA we have


(g),p (yp = ,I (g)z,p (g)y,pI dz
I p(() n Bp(y) I JB, (x) nB (y)

< || t I(g) 'p g (z)| dz
\Bp(x) n Bp(y)| Bp JBp(x)
1 [
+ 1 g() (g),p dj

< c I (g)x,p g(z)| dz + J g(z) (g)y,p dz
1| t JBp (x) |Bp| j )

< c25pA1/2.

Using the two previous inequalities, we have


Ig(x) g(y)l < c26A 1p C26A/2 Ix









for ~-a.e. x, y R C B2 (a). Define A c= c2346, so that for L-a.e. x,y E Rx we
have

(x) u(y)| (g(x) + l x) (g(y) + y)

< 9g(x) gy) + I1 (x y)

< (026 + y11. I 1

Thus there exists a Lipschitz function u: Rx R so that u =u "-a.e. on RA.
By the standard Lipschitz extension theorem (e.g., see section 3.1 of [11]), we may
extend u to a Lipschitz mapping v: B2r(a) R such that

v = u ~-a.e. on RA and sup IVv 11 sup |Vgl < 326
B2r(a) B2,(a)

Step 3. With the above choice of A and by setting

1
3= )(4r,li,a) and 6
8(n + 1)'

we estimate the size of {u / v} as follows: Since u = v ~-a.e. on RX C B2,(a), we
have from step 1 and definitions above that


~(B2,(a) n {u $ v}) _< '(B2,(a) \ R')

< -5B4r 4(4r, 1,a) < ci 'l'(4r, 1, a)1-45. (3 9)
A

Step 4. We need the following
Claim. There exists an r E [', r] so that

S u v| dS < 5 u v dx
JaBF(a) r JB (a)

and


|u (U)a,r l11 (x a) dS < 5 j (u)a,r (x a) dx
/ 5(









are 3. 7. ,/ .: iu ilai *n -];/
Indeed, for any function f E L1(Q), C'!. Iv. !,,v's inequality and Fubini's

theorem gives


{se[,r]: f fdS>5j f dx
fB, T B, f d
1 r
< 5 1 f dSds <
-r \ J| f /2QB


r
5 f,


Ij j If dx
If I dx B,


Thus


Next we let


rs ['r]: f I dS < 5 If dx >'
2 JaB, r JB,: | 2



A s [r, r] u v dS < u
I 2 'B, r B^


and


B s [r, r] u -(u)a,r (x a) dS

< j (u),, -
r B,

Since u e BV(Q) and v E C1',(Q) we note that the functions u -

11 (x a) are both in L1(Q), so that the above argument holds.


3
IAI > 3-r
to


l11 (x a)| dx .

- v and u (u)a,r

Thus we have


3
and |BI > r.
to


However, A, B C [', r], so their intersection must be non-empty. Hence, there is an

r [j, r] which satisfies both the desired inequalities simultaneously as claimed.
For this choice of r, we have

l u v| dS < 5 u v dx
jaB(a) r JB(a) (3-10)
5
< IU vIL(B,(),, (B,(a) n {u u v}).
r


v dx}









For r Ilf ll, < c7 and N(4r, 1i, a) < c28 (the latter being small), Proposition 3.2 and
the estimate (3-9), we have


r5 I
r


as (1 46)."+
that


v||- OL (B,(a) n {u / v})

< (" (B2r(a) n {u / v))1/nLn (B,(a) n {u / v})
C/
< (~ (B2,(a) n {u u ))#

r
C- by c ho (4,i a)1-4v < c (.,T r(4, e, 3a)1+ on

1 + by our choice of J above. Thus, from (3 10), we conclude


(3-11)


SBia)u v dS < c., +(4r, 11, a)1+
By Lemma 3.5 and the preceding inequality, we have
By Lemma 3.5 and the preceding inequality, we have


JB.,(a)


IDu+ f IVul dx + IV(u- w)12 d
SJB,(a)n{|Vu|>1} JB,(a)n{|Vu|
< cs I8 vI dS + c9 /1+26
J9Br(a)
< cs .,i +I(4r, li, a)l + ( .,I I(4r, 11, a)l (+ )

< C30r' [)(4r, 1, la)1+2 + 4(4r, /, a)l+ ] ,1


for any 0 < u < 1/2.
Step 5. Note that Ia + b 2 < 2 [1|a2 + lb|2] for any a, b E R. For any 12 E R, we
can estimate


S|Vu -122 dx<2 | V(u- w)2+ w-2 dx,
y t (a)ing{Vu< } ,) (a)n{aVul
by taking a = V(u w) and b = Vw 12-









Using the last two estimates above, we have

Br I(Dr, 12, a) < D'u + j IVu| dx
JBr( a) JB (a)n{|VuI> 1}
+ w)2 V+ IV 12 2 dx
JBur(a)n{\Vu\ < 2c30n [)(4r, 11, a)'+1 + 1(4r, l1,a) (3-12)

+ C36 /2 dx


We now use Proposition 3.1 for the gradient of w to estimate the last term
above. Let w(x) = w(x) (u)a,r 11 (x a) and note the following:

Aw = Aw =f on B,(a),

w = v3 (U)a,r 11 (x a) on OB,(a),

for r [], r] chosen above; and

Vw(x) Vw(y) = Vw(x) Vw(y),

for any x, y E Bf(a). Hence, we may apply Proposition 3.1 to w, yielding

IVw(x) Vw(y)| Vw(x) Vw(y)|
sup 1/2 sup 1-2
x,yEB,/4(a) I X- y\1 x,yEBF(a) IX 1/2

< C5 [n+1/2 Ia) (U)a,r 1 (x a)| dS + r1/2 |IL1
rn+2 JBF(a)

To complete this estimate, we verify the following claim, which will be used again
later:
Claim. For r, chosen above, we have


IB If 1I3 ()a,r, 11 (x a)I dS < c344)(4r, la)1/2.
1^ (9B)









Indeed, we estimate as follows.

1 F I U -(U)a,r 11 (x a) dS


< | Iv) v dS + B v u dS
IBfI QBf(a) J9BF(a)


1
S [I1 + 12+ ].
|BfI

From Lemma 3.2, we may estimate 11 by

I < (cr1"-) (r1+26) < ( I(4r, i, a)l 1)

For I2, we have by (3-11) that


12 := IU- v dS < c.,i +(4r, 11,a)1'+.
QBf (a)

Finally, by our choice of r and Poincare's inequality, we have for 13


13 <- I \U (U)a,r ( x a) dx < 5c | Du l|.
r B, (a) B, (a)

Putting these estimates together, we have

1
[II +12 + 13
|BrD
< "rn(4r, 1, a)l4+ 1) + rn1(4r, l1, a)+' + IDu I I
JB,(a)
1 2"c
< c(4r,la)2c(4r, l1, a)+2 '+ I|Du +I
r J B(a)
< c32(4r, 11, a) + C Du 11.

Recall that from (3-8) above, we have


c Du 11 < c')(r, 1, a)1/2
rn B,-(a)









Whence,

1 c f
l vPa (U)a,r 11 (x a) I dS < C32(4r 11, a) + c IDu 1i
B' J9Br(a) r Br, (a)
< C324(4r, 11, a) + c')(r, 11, a)1/2 < C34(r, 11, a)1/2

as desired, and the claim is proved.
Since BR/2 < I B, it follows that

|Vw(x) Vw(y)
sup 12
x,yEB,/4(a) I J 1/

SC5 r+1/2 U (U)a,r 1 (x a) I dS+ r1/2 L
rn+2 'JBf(a)

< C37 [-1/2( 1r, ,a)1/2 + r1/2 IlIIL ] ]

By defining 12 = Vw(a) and taking w < -, we may now establish a desirable
estimate for

I |Vw 12 dx

= Vw((x)- Vw(a) 2 dx

J Br (a)
< c(rw)') [w(4r, li,a) + T2 Ifl + 2rw(4r, l,a)1/2 lfllL]


< 39(rw7)n [w,(4r, 1l, a) + r2 If 2]

by Cauchy's inequality. From (3-12), we now see that


IB, 1 (wr, 2, a) < C35rn( (4r, 11, a)1 () + c36 Vw -2 dx
C I(4r, ,aJB (a)) +2 f
< c *I.(4r, l, a)'+ + c(ra)" [LA(4r, l, a) + r2 L].









Dividing by IB, = c(rw)", we conclude that


4(wr, 12, a) < C40o-n()(4r, 11, a)1 1 + C41 I, ))(4 C422 a)+ C4 Il

By choosing w < 1/4 small enough so that c41w0 < 1/4 and restricting

)(4r, li, a) so that c40U -n'((4r, 1, a)4(n+l) < 1/4, we have the desired decay estimate

D (wr, 12, a) < (4r, la) + (4r, 11,a) + ( 1

(4r, l, a)+ c42r2 ll

Step 6. It remains to verify that


Il1 12
To such ends, define

h(x) = w(x) (u),,, 11 (x a)

and note that
Ah = Aw = f in B,(a),

h = v (u)a,~ 1 (x a) on 9B,(a).

We may write h = hi + h2 as the sum of a harmonic function and one with zero

boundary data as follows:

Ah = 0, AAh2 = in Bf(a),
and
hi = vp (U)a,r 11 (x a), h2 = 0 on OB, (a).

From the gradient estimate for harmonic functions, we have

Vhi(x)- J hlv dS,
l BfI | 9Bf(x)

where v is the outward pointing unit normal vector [15]. Thus

|Vhi(x)| < t J |Ihi dS.
BfI J9B(x)








Moreover, for h2, we have that

IVh2 I C31r llfl|L

Recalling that Vw(a) 12, we have


= h(a)

< IVhI(a) + IVh2(a)

< B (f Ihll dS + C31 IIfL

1 I '(
f/ 13 (U)a,r 1 (x

< C34 4(4r, l, a)1/2 + C31r I L ,


- a)I dS + 31r IIfllL


by the claim in step 5. Hence the proposition is proved. E
Theorem 3.1 is now proved by iterating the decay estimate.

Proof. (Theorem 3.1) We will use Proposition 3.3 iteratively. To initialize the
inductive argument assume that

l1r
/ I\Du IiI < co,
| JBr (a)

for some 11 E R" with I1 1 < 1 4p and for any r with r |f |ll- < K. co will be fixed
later. For each x c Br/2(a), we have


4(r/2, 11, x)
S2" B()n{ l}
|Br, B(a)nfjvul

IVu 1112 + B()Vu + B ) ID

2 (BlI, a) 2n (r, 11, a < C43 (a)


so Proposition 3.3 may be applied. For the last inequality, we check each term as
follows: For the part with |VuI < 1, we have

IVu 1112 Vu 11I 1Vu 111 < IVu 111 (IVu + 111<) < 2 IVu li .


I1 121









For the part with IVu| > 1, it suffices to show that there exists a c > 0 so that

IVu < c Vu 11. In fact

IVul IVl IlVu
IVu 1 |Vu 11 Vu- (- 4p)'

which is bounded, as the function constant is decreasing. Thus we have
t--constant


I IVu 11+2 +

S2JB(a)nfnvul< I
Jr(a)r{i\Vu\

|Vu| + |Db|
JB,(a)n {\Vu >1} I+ JBr Du
7u lI| + c \Vu I + |B|I D
JB,(a)n{\Vu>l} J B,

=f IDu 11|
JBr


as claimed.

For the inductive step, choose eo so that c4360 < e and restrict r so that

C18r2 If I200 < c/2. Furthermore, we assume that Ilj-1 < 1 2/ and

'^ j-1 j--1 j-i-142 I
((4)j-1 /, ()J (X) (2) C c44r
i 1
(3-13)

forj 2,..., k. We need to show


( 2l k,Ir) < and Ik <1- 2

to apply Proposition 3.3 and continue the inductive step. Taking w < 1/4, we have
for all k


where it can be

r |f fL- < C/2,


-4


k-1 i-1 < k-2
z(2)-1 <) (yk
(
i 1
shown by induction on k that c45

we have from (3-13)

1 2, k, 2 1
',l,.r)

ltk/2
-1) < c45

v2 By further restricting



SC45 2 C442 ll C.
+ / ca ar Lf _<









By Proposition 3.3 and the inductive assumption we see that
k-i
I k < Ij+ l + 1111
j=1
k-i [ (()l f ) rf1/2(()jr
-- C19I) 2 l, x + C20 r L"
j=1

+1 4- 4
k-1 j-1 j/2 1/2
< C19 2' + C45 C44L2 /1f1

k-1
+C2o0r f LO )Z( + 1- 4/
j=1

k< 1 [()J2 ( 1/2 12 ( /4 C12 L
"C19E 2145 C44 r11/11

j=1 1
k-1
+C20r f L Z4 + 1+1-4/
j 1
A) X1/2
< c() 2,1,x) + c'r I fL|| + c"r IIfl L + 1 -4

-c [4) (1, .) + r || f ||,,] +1 4p,

where the last inequality follows by bounding each term by the appropriate

geometric series. Hence, by restricting eo and r |If |ll again, we can estimate that

I1k\ < 1 2p. Therefore, we may apply Proposition 3.3 to show

S(( ) ) ( \ k 1 (2k+l)/2
2, Ik+1, X ) <, x + C r2 IIL-

where Ilk+1| < 1- 2p.

Whence

lim ) k+1, x 0
k->oo (4 2
uniformly for all x E Br/2(a). Hence


lim X (x D + pVu dx = 0. (3-14)
p-o |Bp(x)| [J() JBVn{\>u \}









It follows from Lemma 1 in section 1.6 [11] that

|Dul (Br/2(a)) = 0.

Moreover, since

lim I Vu dx 0
p-, IBp(Xx)l JBn{vul>1i}
we conclude that

L' (B,/2(a) n {Vul > 1}) 0.

Thus is follows that Vul < 1, and by the first variation, -Au = f on Br/2(a) as

desired. O

3.3 Regularity of the Elastic Region

We prove Theorem 3.2. Assume that u is a minimizer and that the set

E = {|Vul < 1} has positive Lebesgue measure.

Proof. (Theorem 3.2) Since u E BV, its derivative Du is a Radon measure

that can be decomposed into singular and absolutely continuous parts with respect

to Lebesgue measure, D"u and Vu, respectively. It follows by Proposition 6.8 [23]

that ID"ul is singular with respect to Lebesgue measure as well. Thus, by Theorem

7.13 [23], we have from (3-14)

1 P
lim 1 Df ul -0,
r- I B | B(x)

for "-a.e x E Furthermore, since Vu e LI(Q), we have

lim t Vu(y)- Vu(x) dy 0,
r-0 Br J(x)

for L-a.e. x E E, by the Lebesgue point theorem.

From E, remove points at which either of the above fails to hold; call the

resulting set E. Thus we have L(E \ E) = 0, IVu < 1 on E and both the above

limits hold for each point in E. For each fixed x c E, there exists some p, > 0 such









that

Vul < 1 2p1.

Letting I Vu(x), we see from above that


t j Du -l < Vu(y) -l dy + j Dsuu 0.
Thus, r given above, there exists r > 0 so that
Thus, for co given above, there exists rx > 0 so that


I B I


IDu l < o0.


Therefore, Theorem 3.1 gives that

ID'su (B,,(x)) 0 and |Vul < 1 px on B, (x),

with u CE C('(Br.(x)). Hence Br,(x) C E, showing that E is an open set with the

desired properties.

O

Even with the safe-load condition (which guarantees the existence of a

minimizer), it is not clear whether the set E has positive measure or not. We next

prove Theorems 3.3 and 3.4, which are simple comparison arguments, discussed for

example by Hardt and Kinderlehrer [18].


Proof. (Theorems 3.3 and 3.4) To get a contradiction, assume that

({IVul < 1}) 0.


Under this assumption, F(Du) = IDul

can choose some v E BV(Q) with van


, a.e. For a given boundary data p we
p such that


I' \\BV(Q) c( ) < C I lLI(an)








by the trace theorem for BV functions ([16]). By the minimality of u, we have

J F(Du) < F(Dv) f(v- u) dx

< c(A) |0111(a) + CD(Q) If L| (Du| |Dv),

where CD is the smallest constant such that I||I li(Q) < CD fJ ID(I, for all e
BV(Q) such that (n = 0. Assume that ||fL|| < 16 for some 6 > 0. Then

1 IDu < || F(Du)

< C() ||||L(a) + L IDul + CD ILf L IL'h()

It follows that

| Il|Dul < [CD Ifl|L| + C()] IllL(a0)
2 2 J"

If II(IIL1(,) is sufficiently small, this is not possible. Hence, for small enough
norms, we have a contradiction. Thus {|Vul < 1} has positive measure and
Theorem 3.2 applies. The proof of Theorem 3.4 follows similarly using v = 0 as a
comparison.














CHAPTER 4
CONCLUSION

The main result of C'!i pter 2, Theorem 2.1, shows that the variational

integral f F(m), for a vector-valued measure m, can be defined ii i ii ,lly" by the

Fenchel transform. An important consequence of the technique is that the lower

semi-continuity of the functional follows immediately from well-known properties of

the Fenchel transform from complex analysis. In our approach, we decomposed the

measure with respect to Lebesgue measure: m = m' + m", where ma is absolutely

continuous to Lebesgue measure and m" is mutually singular to Lebesgue measure.

One question of interest that was not explored is whether Lebesgue measure is the

best choice for the base of this decomposition.

To establish the duality needed for the Fenchel transform, we considered the

space of continuous bounded functions, CB, which induces a topology (the weak

topology) onto the space of bounded vector-valued measures, M. Is CB the best

choice to establish the required duality? Another possible candidate is the Sobolev

space W1'"(Q, R, ); with this choice, the weak topology on AM is metrizable.

In C'! lpter 3, we established a partial regularity result for the plasticity

problem. The basis of our technique is the decay estimate for the :; .. -- given in

Proposition 3.3. Hardt and Tonegawa [20] give a partial regularity result for a weak

solution to the evolution problem

au
-t divFp(Vu),


where u = f on 2 x {0} and u = g on 02 x (0, T). Their result is similar to

Theorem 3.1; however, the decay method they used is dependent upon the space









variable having dimension n = 1 or 2 [20]. One extension of their result that to

consider is to establish this result for the general n-dimensional space variable.

Another extension to explore is to obtain a partial regularity result for the

problem
Bu 1
au= div,F,(Vu) + u -),
at 2

where I is given and depends on the space variable x. A weak solution to this

problem minimizes

F(Du) + ( I(x))2 (41)

over BV(Q). An important application for this problem is image restoration.

In this case F may be the function given in (1-3) and I is the observed image.

The main distinction between this problem and the one considered by Hardt and

Tonegawa[20] is the dependence of (4-1) on the function u and the space variable

x.














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BIOGRAPHICAL SKETCH

I received a Bachelor of Arts degree in 1994 from Ohio Wesleyan University.

At OWU, I n1 i, Pred in mathematics and earned minors in economic management

and phi~. phi,-. In 1997, I received a Master of Arts degree from Bowling Green

State University in mathematics. I entered the doctoral program at the University

of Florida in 1998; I began work with Dr. C'!I in in 2001.