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A REMARK ON AN INTEGRAL CHARACTERIZATION OF THE DUAL OF BV NICOLAFUSCO 7 1 DipartimentodiMatematicaeApplicazioniRenatoCaccioppoli,Universita`degliStudidi 0 NapoliFedericoII,Napoli,Italia 2 n a J DANIELSPECTOR 3 DepartmentofAppliedMathematics,NationalChiaoTungUniversity,Hsinchu,Taiwan 1 ] A ABSTRACT. Inthispaper,weshowhowunderthecontinuumhypothesisonecan obtainanintegralrepresentationforelementsofthetopologicaldualofthespace F offunctionsofboundedvariationintermsofLebesgueandKolmogorov-Burkill . integrals. h t a m 1. INTRODUCTIONANDMAINRESULTS [ Let Ω Rd be open and denote by BV(Ω) the set of functions of bounded ⊂ 1 variationinΩ,thatis,thosefunctionsu L1(Ω)withfinitetotalvariation: ∈ v 4 (1.1) Du(Ω):= sup udivΦ<+ . | | ˆ ∞ 7 Φ∈Cc1(Ω;Rd), Ω 5 kΦkC0(Ω;Rd)≤1 3 Whenequippedwiththenorm 0 1. kukBV(Ω) :=kukL1(Ω)+|Du|(Ω), 0 BV(Ω) is a Banach space, though in contrast to the Sobolev case, smooth com- 7 pactlysupportedfunctionsfailtobedenseinthenormtopology,noristhespace 1 even separable. Its importance then lies in the fact that it is in some sense the : v naturalspaceinavarietyofproblemsinthecalculusofvariationswhereenergies i exhibitlinearboundsinthegradientofthefield-anotableexampleisinthestudy X ofminimalsurfaces(see[4]forthesystematicstudyofthespaceinthiscontext). r Akeytoolinitsstudyisthecompactnessofboundedsetswithrespecttothe a weak-starconvergenceofmeasuresprovidedbytheBanach-Alaoglutheorem,as BV(Ω)canbeembeddedintheproductspaceofintegrablefunctionscrossedwith Rd-valuedfiniteRadonmeasuresL1(Ω) Mb(Ω;Rd).Morethanthis,BV(Ω)isit- × selfadualspace(aresultwedescribeinSection2inmoredetail)forwhichone hasanintegralrepresentationofthedualitypairing. However,aproblemwhich hasnotbeenclearlyexplainedisthatofgivinganintegralcharacterizationofits dual. Wewerecurioustoseeifthiscouldbedone,astheliteratureonthespace itselfhasonlypartialknownresults.Theproblemcanbedatedtoatleastthe1977 paperofMeyersandZiemer[8]onPoincare´-Wirtingerinequalitiesthatprovidesa characterizationofthepositivemeasuresinthedualofBV(Ω),whilesuchaques- tionisraisedexplicitlyinthe1984AMSsummermeetingonGeometricMeasure E-mailaddresses:[email protected], [email protected]. 1 2 TheoryandtheCalculusofVariations(see[2], Problem7.4onp.458). Itisagain statedinthe1998paperofDePauw[3],whogivesanintegralrepresentationfor thedualofthespaceofspecialfunctionsofboundedvariation.Morerecently,the problemhasbeendiscussedinthepapersofPhucandTorres[9]andTorres[10]. ItturnsoutthatonecangiveanintegralrepresentationofthedualofBV(Ω), andthatthesolutionexistedmoreorlessbeforeanyreferencetosuchaproblem. Indeed,inthepapers[6,7],assumingthecontinuumhypothesis,Mauldincharac- terizedthedualofspacesoffiniteRadonmeasuresinthecasethesetofallRadon measuresonthespacehascardinalityatmost2ℵ0. Asaconsequenceofhisresult andtheHahn-Banachtheorem,onecaneasilygetthefollowingintegralrepresen- tation for the dual of the BV(Ω). To this end, let us recall that given a function u BV(Ω),themeasurederivativeDucanbewrittenas ∈ Du= u d+Dju+Dcu, ∇ L where uisabsolutelycontinuouswithrespecttotheLebesguemeasure d,Dju= ∇ L νu(u+ u−) d−1 JuwithJua(d 1)-rectifiableset,andDcu=Dsu Djufor − H − − DsutheportionofDuwhichissingularwithrespecttotheLebesguemeasure. Theorem1.1. LetΩ Rd beopenandL (BV(Ω))0. UnderZFCsettheoryandthe ⊂ ∈ ContinuumHypothesis, thereexistsg L∞(Ω), G0 L∞(Ω;Rd), an d−1 bounded ∈ ∈ H andmeasurablefunctionG1 :Ω RdandΨ: (Ω) Rabounded,Borelsetfunction → B → forwhich L(u)= gudx+ G udx+ G dDju+(K) Ψ dDcu ˆ ˆ 0·∇ ˆ 1· ˆ · Ω Ω Ju for all u BV(Ω). Here, the last integral on the right hand side is understood in the ∈ Kolmogorov-Burkill sense. Conversely, any such integral functional is in the dual of BV(Ω). HereweobservethattheintegralrepresentationisintermsofbothLebesgue andKolmogorov-Burkillintegrals. Werecallthatgivenasetfunctionψ : (Ω) B → R and a countably additive set function µ : (Ω) [ , ], the Kolmogorov- B → −∞ ∞ Burkill integral is defined as follows. We say that ψ is Kolmogorov-Burkill inte- grablewithrespecttoµifthereexistsanumberI Rsuchthatforeveryε > 0 ∈ thereisafinitepartitionofΩintoBorelsetsDsuchthatifD0isanyrefinementof D, ψ(B)µ(B) I <ε. (cid:12) − (cid:12) (cid:12)(cid:12)BX∈D0 (cid:12)(cid:12) AsdiscussedbyKolmogoro(cid:12)vin[5], thisnotion(cid:12)ofintegrationencompassesboth (cid:12) (cid:12) LebesgueandRiemannintegration. Indeed,theKolmogorovintegralgeneralizes the idea of Leibniz on areas as set functions, a sort of Riemann integral for the set function ψ µ. Notice that while µ is countably additive, ψ satisfies no such · property. Moreover, for any fixed µ M (Ω), the Kolmogorov-Burkill integral b ∈ projectsontoaLebesgueintegral. Infact,ifψ : (Ω) RisKolmogorov-Burkill B → integral with respect to µ, then there exists gµ L∞(Ω, (Ω), µ) such that for ∈ B | | everyf L1(Ω, (Ω), µ) ∈ B | | (K) ψd(fµ)= fg dµ. ˆ ˆ µ Ω Moregenerally, sucharepresentationformulaholdswhenever(Ω, (Ω),µ)isan B almostdecomposablemeasurespace.Thisisanimmediateconsequenceofaresult byDePauw(seeLemma3.4in[3]). 3 Althoughwearenotabletoprovethatthecontinuumhypothesisisnecessary to give the integral representation in Theorem 1.1, Theorem 3.14 in [3] suggests thatitmightbeso.However,itisinterestingtoobservethatinonedimensionone cancharacterizethedualofSBV(a,b)withoutthecontinuumhypothesis.Indeed, given L (SBV(a,b))0, we can separate its action on u,u0 L1(Ω) and Dju = ∈ ∈ Pfunx∈ctJiuonus(xg+,g)−0aun(xd−a)f.uTnhcitsiodnecgo1u:p(lain,bg)i→mpRliessutchhatththaterseupexx∈is(tas,bt)w|go1(bxo)u|n<de+d∞B,ofroerl which L(u)=ˆ gudx+ˆ g0u0 dx+ g1(x)(u(x+)−u(x−)), Ω Ω xX∈Ju forallu SBV(a,b). ∈ 2. PROOFSOFTHEMAINRESULTS IntheintroductionwementionthatBV(Ω)isadualspace. Wehereprovidea prooffortheconvenienceofthereader.Ifonedefinesthespaceofdistributions d ∂ V(Ω):={T ∈D0(Ω):T =ϕ0+ ∂x ϕj :ϕj ∈C0(Ω)} j j=1 X equippedwiththenorm d g := ϕ + ϕ , k kV(Ω) k 0k∞ k jk∞ j=1 X thenweshow Proposition2.1. LetL (V(Ω))0.ThenthereexistsuL BV(Ω)suchthat ∈ ∈ L(T)= u ϕ dx+ Φ dDu ˆ L 0 ˆ · L Ω Ω foreveryT V(Ω)suchthat ∈ T =ϕ +divΦ. 0 Proof. WerecallthatL (V(Ω))0means ∈ L(αT +βS)=αL(T)+βL(S), L(T) C T , | |≤ k kV(Ω) forallα,β R,S,T V(Ω). LetL (V(Ω))0begiven. ThentakingT =ϕwhere ∈ ∈ ∈ ϕ C (Ω)wehave 0 ∈ L(ϕ) C ϕ , | |≤ k k∞ sothatbytheRieszrepresentationtheoremwehaveL M (Ω)and b ∈ L(ϕ)= ϕdµ . ˆ L Ω ThenforeveryΦ∈Cc1(Ω;Rd),wehavebycontinuityandthisrepresentation L(divΦ) C Φ | |≤ k k∞ Thus,µ M (Ω)issuchthatitsdistributionalderivatives ∂µL M (Ω),sothat L ∈ b ∂xj ∈ b µL =uL BV(Ω). (cid:3) ∈ InordertoproveTheorem1.1,werequirearesultofMauldin. Thenextstate- mentisaspecialcaseofTheorem10in[7]. 4 Theorem2.2. LetΣbeaσ-algebrainΩ Rdsuchthatthesetofallcountablyadditive ⊂ real-valuedmeasuresonΣhascardinalityatmost2ℵ0.Assumethecontinuumhypothesis, 2ℵ0 = 1.ThenforeveryL (Mb(Ω,Σ))0thereexistsaboundedsetfunctionψ:Σ R ℵ ∈ → suchthat L(µ)=(K) ψdµ ˆ foreveryµ Mb(Ω,Σ). ∈ WiththistheoremwecannowgivetheproofofTheorem1.1. Proof. FirstletusobservethatwemayidentifyBV(Ω)withaclosedsubspaceof L1(Ω) Mb(Ω;Rd). Indeed, the space of distributional gradients is curl-free, in × thesenseofdistributions,andsoisclosedwithrespecttosequentialconvergence in L1(Ω), and so in particular it is closed with respect to the norm topology on BV(Ω).Thus,foranyL (BV(Ω))0,wemaybytheHahn-Banachtheoremextend ∈ LtoafunctionalL (L1(Ω) Mb(Ω;Rd))0suchthat ∈ × L(u)=L(u,Du) for all u BV(Ω). Now, for any L (L1(Ω) Mb(Ω;Rd))0, we may separate ∈ ∈ × its action on L1(Ω) and Mb(Ω;Rd). Thus, by the classical Riesz representation theoremwehave L(f,µ)= fgdx+L˜(µ) ˆ Ω for some g L∞(Ω), L˜ (Mb(Ω;Rd))0, and all f L1(Ω) and µ Mb(Ω;Rd). ∈ ∈ ∈ ∈ We can now invoke Theorem 2.2, since the cardinality of the space of countably additivereal-valuedmeasureson (Ω)is2ℵ0. B To see this, it is enough to show that a finite non-negative Borel measure is determined by its values on open balls B(x,r) Ω where x Qd and r Q+. ⊂ ∈ ∈ Indeed,iftwofinitenon-negativeBorelmeasuresµ,νcoincideonsuchballs,then byapproximationtheyalsocoincideonanyclosedballB(x,r) Ωforx Ωand ⊂ ∈ r>0.ItnowsufficestoshowthatµandνcoincideonallopensubsetsofΩ,since byapproximationtheywillcoincideonanyBorelsubsetofΩ. Thus,letU Ωbe ⊂ open.AnapplicationoftheVitali-Besicovitchcoveringtheorem(see,forinstance, Theorem2.19in[1])tothemeasureµ+νyieldsasequenceofdisjointclosedballs B(x ,r ) U suchthat n n ⊂ (µ+ν) U B(x ,r ) =0. n n n \∪ Thus, (cid:0) (cid:1) µ U B(x ,r ) =ν U B(x ,r ) =0, n n n n n n \∪ \∪ sothat (cid:0) (cid:1) (cid:0) (cid:1) µ(U)=µ B(x ,r ) =ν B(x ,r ) =ν(U). n n n n n n ∪ ∪ Thusthereexistbound(cid:0)edBorelfunc(cid:1)tions(cid:0)ψj : (Ω) (cid:1)R,j =1...dsuchthat B → d L˜(µ)= (K) ψ dµ ˆ j j j=1 X foranyµ Mb(Ω;Rd),wherewehaveµ=(µ1,µ2,...,µd).DefineΨ:=(ψ1,ψ2,...,ψd), ∈ sothatwemaywritetheprecedingmorecompactlyas L(f,µ)= fgdx+(K) Ψ dµ. ˆ ˆ · Ω 5 NowobservethattherestrictionofL˜ : L1(Ω;Rd) Rislinearandcontinuous, → andthereforeagainbytheclassicalRieszrepresentationtheoremthereexistsG 0 ∈ L∞(Ω;Rd)suchthat (K) Ψ d(F d)= G Fdx ˆ · L ˆ 0· Ω forallF L1(Ω;Rd).WecantreatinasimilarmannertherestrictionL˜ :L1(Ω, (Ω), d−1;Rd) ∈ B H → R, since (Ω, (Ω), d−1) is an almost decomposable measure space, and so by B H Lemma3.4in[3]themap :L∞(Ω, (Ω), d−1) (L1(Ω, (Ω), d−1))0 Y B H → B H givenby Y(g)(f)=ˆ gf dHd−1 Ω is onto. Therefore there exists an d−1 bounded and measurable function G1 : H Ω Rdsuchthat → (K)ˆ Ψ·d(FHd−1)=ˆ G1·F dHd−1. Ω forallF L1(Ω, (Ω), d−1). ∈ B H Thenrestrictingtherepresentationtou BV(Ω),alongwiththedecomposition ∈ Du= u d+Dju+Dcu, ∇ L andlinearityoftheKolmogorov-Burkillintegralwithrespecttothemeasure,we finallydeducethat L(u)= gudx+ G udx+ G dDju+(K) Ψ dDcu. ˆ ˆ 0·∇ ˆ 1· ˆ · Ω Ω Ω (cid:3) ACKNOWLEDGEMENTS The authors would like to thank Cindy Chen for her assistance in obtaining some of the articles utilized in the researching of this paper. N.F. supported by PRIN2015PA5MP7’CalcolodelleVariazioni‘,D.S.supportedbytheTaiwanMin- istry of Science and Technology under research grant 105-2115-M-009-004-MY2. Part of this work was written while N.F. was visiting NCTU with support from theTaiwanMinistryofScienceandTechnologythroughtheMathematicsResearch PromotionCenter. 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