An integral representation result is obtained for the relaxation of a class of energy functionals depending
on two vector fields with different behaviors

Ana Margarida, Ribeiro

Carita, Graça

Elvira, Zappale

Repositório Científico da Universidade de Évora

Relaxation for some integral functionals in W 1;pw  LqwGraca Carita, Ana Margarida Ribeiroy, Elvira ZappalezNovember 23, 2010AbstractAn integral representation result is obtained for the relaxation of a class of energy functionals de-pending on two vector elds with dierent behaviors.Keywords: Relaxation, quasiconvexity-convexity.MSC2000 classication: 49J45, 74Q051 IntroductionMinimization of energies depending on two independent vector elds arise frequently to model several phe-nomena. This is the case of thermochemical equilibria for multiphase solids (see [4]) or TV (total variation)image decomposition models (see [8], [9]). In both cases, the functionals to minimize have an integral formwith densities depending on the total variation of a vector eld u and on another independent vector eldv. However, the natural space setting is dierent in each case. For the rst one, u is a Sobolev function,whereas in the image decomposition models the natural space for u is BV , the space of bounded variationfunctions.In this paper we consider functionals of the formZf(x; u(x);ru(x); v(x)) dx; for u 2W 1;p(;Rd) and v 2 Lq(;Rm) (p; q  1):As well known for the case with no x and u dependence, sequential weak lower semicontinuity of theprevious functional is related to convexity conditions of the density f (cf. [4], [7]). More precisely, under con-venient growth and coercivity conditions, a joint quasiconvexity of f on the gradient variable and convexityon the v variable, here called quasiconvexity-convexity, cf. Denition 1.3 below, is a necessary and sucientcondition for weak lower semicontinuity. In the lack of such condition it is of interest to characterize therelaxed functional of the original one. In this paper we provide a characterization for the relaxed energyof the functional above for the weak convergence in W 1;p and the weak convergence in Lq, generalizing aresult due to Fonseca, Kinderlehrer and Pedregal cf. [4] where no dependence on x and u was considered.Moreover, this is a rst step to deal with the analogous relaxation problem in a BV  Lqw setting, as it isthe case for image decomposition models.We introduce next some notation and we establish our main result. Let  be an open bounded set ofRN and let A () denote the family of all open subsets of :Let 1  p; q <1 and let F :W 1;p  ;Rd Lq (;Rm)A ()! R be dened byF (u; v;A) :=ZAf (x; u (x) ;ru (x) ; v (x)) dx (1.1)CIMA-UE, Departamento de Matematica, Universidade de Evora, Rua Rom~ao Ramalho, 59 7000 671 Evora, Portugale-mail: gcarita@uevora.ptyDepartamento de Matematica and CMA, Faculdade de Cie^ncias e Tecnologia - Universidade Nova de Lisboa, 2829-516Caparica, Portugal e-mail: amfr@fct.unl.ptzD.I.I.M.A., Universita' degli Studi di Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA) Italy e-mail: ezappale@unisa.it1where f :  Rd  RdN  Rm ! R is a Caratheodory function satisfying0  f (x; u; ; b)  C (1 + jujp + jjp + jbjq) (1.2)for a.e. x 2 ; for every (u; ; b) 2 Rd  RdN  Rm and for some C > 0:Consider the relaxed localized energy of (1:1) given byF (u; v;A) := inflim infn!1ZAf (x; un (x) ;run (x) ; vn (x)) dx : un * u in W 1;p A;Rd; vn * v in Lq (A;Rm):Our goal is to nd an integral representation for F : PreciselyTheorem 1.1 Let 1  p; q <1 and assume that f :  Rd  RdN  Rm ! R is Caratheodory satisfyingthe growth condition (1:2).Then for every u 2W 1;p  ;Rd ; v 2 Lq (;Rm) and A 2 A () we haveF (u; v;A) =ZAQCf (x; u (x) ;ru (x) ; v (x)) dx;where QCf stands for the quasiconvex-convex envelope of f with respect to the last two variables (see (1.5)and (1.6)).Remark 1.2 Assuming that f satises the coercivity condition1C(jjp + jbjq)  C  f (x; u; ; b) (1.3)for a.e. x 2 ; for every (u; ; b) 2 Rd  RdN  Rm and for some C > 0 then F is W 1;pw  Lqw lowersemicontinuous.We also emphasize that Theorem 1.1 constitutes an extension of [1, Theorem 1.3] for the case of thedierential operator A = (curl; 0), since, in the present result, we impose two dierent growth assumptionson the A-free elds.Following [4, 7], see also [5], we recall the denition of quasiconvexity-convexity.Denition 1.3 A Borel measurable function f : RdN  Rm ! R is said to be quasiconvex-convex if thereexists a bounded open set D of RN such thatf(; b)  1jDjZDf( +r'(x); b+ (x)) dx; (1.4)for every (; b) 2 RdNRm, for every ' 2W 1;10 D;Rdand for every  2 L1(D;Rm), with RD(x)dx = 0.If f : RdN  Rm ! R is any given Borel measurable function bounded from below, it can be denedthe quasiconvex-convex envelope of f , that is the largest quasiconvex-convex function below f :QCf(; b) := supfg(; b) : g  f; g quasiconvex-convexg: (1.5)Moreover, by Theorem 4.16 in [7]QCf(; b) = inf1jDjZDf( +r'(x); b+ (x)) dx : ' 2W 1;10 (D;Rd);  2 L1(D;Rm);ZD(x) dx = 0;(1.6)D being any bounded open set.Remark 1.4 It can also be showed that if f satises (1:2) for some p; q  1, then in (1:4) and (1:6) thespaces W 1;10 and L1 can be replaced by W 1;p0 and Lq, respectively.22 ResultsAll over this section f :  Rd  RdN  Rm ! R is a Caratheodory function satisfying (1:2) :We start by presenting the following auxiliary lemmas.Lemma 2.1 Let 1  p; q < 1. For any A 2 A () ; u 2 Lp  A;Rd, v 2 Lq (A;Rm) ; and any sequencesfung  Lp A;Rdand fvng  Lq (A;Rm) such that un * u in W 1;p A;Rdand vn * v in Lq (A;Rm)there exist sequences fung  W 1;p A;Rd; fvng  Lq (A;Rm) such that un * u in W 1;p A;Rd; frungis p equi-integrable, RArun (x) dx =RAru (x) dx, vn * v in Lq (A;Rm), fvng is q equi-integrable,RAvn (x) dx =RAv (x) dx andlim infn!1ZAf (x; u (x) ;run (x) ; vn (x)) dx  lim infn!1ZAf (x; un (x) ;run (x) ; vn (x)) dx:Proof. If p; q > 1; the proof is identical to [1, Lemma 3.1], together with the second part of the proof of[1, Theorem 1.3, page 560]. The mentioned lemma gives the construction of the sequences fwng, (curl-free)and fvng while the argument of [1, Theorem 1.3] ensures that the sequence fwng is in fact a sequence ofgradients frung :If p = 1 (q = 1; respectively), then un = un (vn = vn; respectively) because the original sequences arealready equi-integrable.Let L :W 1;p  ;Rd Lq (;Rm)A ()! R be dened byL (u; v;A) := inflim infn!1ZAf (x; u (x) ;run (x) ; vn (x)) dx : frung  Lp A;RdN;run * ru in Lp A;RdN; frung is p  equi-integrable,fvng  Lq (A;Rm) ; vn * v in Lq (A;Rm) , fvng is q   equi-integrable):Lemma 2.2 Let 1  p; q < 1: Then F (u; v; ) and L (u; v; ) are the traces of Radon measures absolutelycontinuous with respect to LNb:Proof. The proof is standard relying on De Giorgi-Letta criterion (see [3]) and we refer to Lemma 3.4 in [1]for more details.Lemma 2.3 Let 1  p; q <1: ThenF (u; v;A) = L (u; v;A)for any u 2W 1;p  ;Rd ; v 2 Lq (;Rm) and A 2 A () :Proof. The result is achieved by double inequality. The proof of L (u; v;A) F (u; v;A) relies on Lemma2:1. Indeed, for fung  W 1;p A;Rdand fvng  Lq (A;Rm) such that un * u in W 1;p A;Rdandvn * v in Lq (A;Rm) we may nd sequences fung  W 1;p A;Rdwith frung p   equi-integrable, andfvng  Lq (A;Rm) ; q   equi-integrable such thatlim infn!1ZAf (x; u (x) ;run (x) ; vn (x)) dx  lim infn!1ZAf (x; un (x) ;run (x) ; vn (x)) dx:Using the denition of L and taking the inmum on the sequences fung and fvng we obtain L (u; v;A) F (u; v;A) :To obtain the converse inequality, consider admissible sequences for L and, using Poincare inequality andthe fundamental theorem of Young measures, construct an admissible sequence for F with the same limitenergy. See the proof of Theorem 1.3 in [1] for more details.We sketch the proof of the main result.Proof of Theorem 1.1. The characterization of F is obtained by double inequality.3To prove thatF (u; v;A) ZAQCf (x; u (x) ;ru (x) ; v (x)) dx; (2.1)for every u 2W 1;p  ;Rd, v 2 Lq (;Rm) and A 2 A () ; we can exploit blow-up techniques developed byFonseca and Muller in [6] and argue as in the proof of [2, Theorem 1.2], introducing the dependence on u inthe density f: We also observe that the same proof of [2, Theorem 1.2] can still be performed replacing thegrowth condition therein by (1:2) : Moreover, the result still holds for q = 1:To prove the converse inequality of (2:1) ; we observe that by Lemma 2.3, it is enough to show thatL (u; v;A) ZAQCf (x; u (x) ;ru (x) ; v (x)) dx (2.2)for every u 2W 1;p  ;Rd, v 2 Lq (;Rm) and A 2 A () :By Lemma 2.2 L (u; v; ) is a measure. In the case p > 1 and q > 1, to prove (2:2) we argue as in thesecond part of the proof of [2, Theorem 1.2], generalizing [2, Theorem 3.1] to the present growth condition,namely, replacing condition iii) therein byjL (u; v;A)j ZAa (x) + C (jru (x)jp + jv (x)jq) dxfor some C > 0 and a 2 L1loc () :If p = 1 and (or) q = 1 the upper bound can be proved arguing as in [1, Theorem 3.6].Acknowledgements. The research of G. Carita was supported by CIMA-UE nanced by FCT (Fundac~aopara a Cie^ncia e Tecnologia). The research of A. M. Ribeiro was partially supported by Fundac~ao paraa Cie^ncia e Tecnologia, Financiamento Base 2010-ISFL/1/297 from FCT/MCTES/PT. The research of E.Zappale was partially supported by D.I.I.M.A. Universita degli studi di Salerno, fondi FARB 2008. E.Zappale also acknowledges the support of Math. Dept. of FCT-UNL and is very grateful for its hospitality.References[1] A. Braides, I. Fonseca, & G. Leoni, A-quasiconvexity: relaxation and homogenization, Control.Optimization and Cal. Var., 5, (2000), 539-577.[2] G. Carita, A. Ribeiro, & E. Zappale, An homogenization result in W 1;p  Lq (submitted).[3] E. De Giorgi, & G. Letta, Une notion generale de convergence faible pour des fonctions croissantesd'ensemble, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4), 4, (1977), 61-99.[4] I. Fonseca, D. Kinderlehrer, & P. Pedregal, Energy functionals depending on elastic strain andchemical composition, Calc. Var. Partial Dierential Equations, 2, (1994), 283-313.[5] I. Fonseca, D. Kinderlehrer, & P. Pedregal, Relaxation in BV L1 of functionals depending onstrain and composition, Lions, Jean-Jacques (ed.) et al. Boundary value problems for partial dierentialequations and applications. Dedicated to Enrico Magenes on the occasion of his 70th birthday. Paris:Masson Res. Notes Appl. Math. , 29, (1993), 113-152.[6] I. Fonseca, & S. Muller, Quasi-convex integrands and lower semicontinuity in L1, SIAM J. Math.Anal., 23, (1992), 1081-1098.[7] H. Le Dret & A. Raoult, Variational convergence for nonlinear shell models with directors and relatedsemicontinuity and relaxation results, Arch. Ration. Mech. Anal. 154, (2000), No.2, 101-134.[8] Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations, University Lec-ture Series, AMS 22 (2001).[9] L. Rudin, S. Osher & E. Fatemi, Nonlinear total variation based noise removal algorithms, PhysicaD 60, No. 1-4, (1992), 259-268.4

Relaxation for some integral functionals in W1;p w × Lq w

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Relaxation for some integral functionals in W1;p w × Lq w

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