Homogenization of supremal functionals

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1 Homogenization of supremal functionals 26th December 2002 Ariela Briani Dipartimento di Matematica Universitá di Pisa Via Buonarroti 2, Pisa, Italy Adriana Garroni Dipartimento di Matematica Universitá La Sapienza P.le Aldo Moro 5,00185 Roma, Italy Francesca Prinari Dipartimento di Matematica Universitá di Pisa Via Buonarroti 2, Pisa, ITALY Abstract. Mathematics Subject Classification (2000): Keywords: 1 Introduction In contrast to classical problems of the calculus of variations, where the minimization of integral energies is considered, the last decade has witnessed a growing interest in problems where one would like to minimize some quantity which does not express a mean property, or whose value can be relevant on sets of arbitrarily small measure and thus it cannot be represented by an integral functional. As an example, part of the literature on such problems was originally motivated by the following mathematical problem: find the best Lipschitz extension in of a function ϕ defined on. This can be clearly expressed in the following variational form inf u=ϕ Du L (). Variants of this problem have been extensively studied by many authers using approximations results and the theory of viscosity solutions (see for instance [1], [?], [?], ecc... and Juutinen thesis [?] and the references therein). On the other hand, in the applications one encounters many optimitation problems whose criteria select solutions which minimize an extremal quantity (e.g., involving the pointwise maximum of an energy density). An example is the problem of modelling the dieletric beakdown for a composite conductor. In [?] such a model has been derived; in particular, in the case of a composite material made of two ogeneous phases the variational principle that one has to consider is the following min ess sup λ(x)du : Du dx = ξ (1.1) 1

2 where λ(x) is a piecewise-constant function taking values α and β (the two phases) and the condition that Du dx = ξ corresponds to assign the average electric field. The idea is that when this minimum reaches a certain threshold the first-failure dielectric breakdown occurs. Asimilar model could also be applied to other physical situation like the polycrystal plasticity problem in the case of anti-plane shear (as formulated, e.g., by [?]). More in general, we can consider the following functionals F (u) = ess sup f(x, Du(x)) (1.2) x where R N and u W 1, (). These functionals have been recently studied using the direct method of the calculus of variations and a partial theory has been developed parallel to the well established theory for integral functionals. In [?], where they have been baptized supremal functionals, a representation theorem has been proved. Subsequently, their semicontinuity has been completely characterized in [?]. The related relaxation problem was studied in [?] using viscosity solutions techniques, thus requiring a continuous dependence on the x variable, the general case being still open. The question whether this class of functionals is stable under Γ-convergence in L arise naturally. It has been first approached in [5] in the case when the functionals do not depend on the gradient, i.e. they are of the form ess sup f(x, u(x)), (1.3) x but the case of functionals of the form (1.2) is still open. In this paper we give a partial answer to this question studying the case of ogenization. Namely, we consider the following sequence of functionals F ε (u, ) := ess sup f( x,du(x)), (1.4) x ε where is a bounded subset of R N, u W 1, () and f is periodic in the first variable. The aim is to replace this highly oscillating functional, as ε goes to zero, with a simpler functional F, the ogenized functional, which captures the relevant features of the sequence F ε. This problem has been considered in [?] in the particular form (1.1) in order to study the macroscopic behaviour of a two-phases composite material for the first failure dielectric breakdown model. The authors use an approximation approach: they prove that the approximation of (1.1) by power law functionals used by material scientists and considered by [?] and [?] in the case of the viscosity solutions approach, is indeed a Γ-limit as p. More precisely, for any fixed ε>0, they consider the sequence of functionals ( F p,ε (u) := λ( x dx) ε )Du(x) p if u W 1,p (), (1.5) + otherwise in L 1 () with λ periodic and 0 <α λ(x) β, and they prove its Γ-convergence (with respect to the L 1 topology) to the supremal functional sup λ( x F ε (u) := ε )Du(x) if u W 1, () (1.6) + otherwise in L 1 (). One can consider the ogenized functional for (1.5), which describes the macroscopic behaviour of the approximating power law materials and by the classical ogenization theory is given by (the power of) an integral functional, Fp, whose energy density fp is ogeneous and is given by the following cell-problem formula ( fp (ξ) := inf λ p (x) Du(x)+ξ p dx : u W 1,p ((0, 1) N ), periodic (0,1). N Thanks to the Γ-convergence of F p,ε as p one has that the functions fp converge, as p,to the function f defined by f (ξ) := min sup λ(x) Du(x)+ξ : u W 1, ((0, 1) N ), periodic. (1.7) 2

3 In view of the application to the computation of bounds for composite materials, in [?], the authors consider f as the relevant energy density for the macroscopic behaviour of the limit problem F ε, but they do not actually prove that it can be obtained directly by ogenizing it. This will be the main result of our paper in a broader context. We will prove the ogenization theorem for supremal functionals of the general form (1.4), under very mild assumptions for the function f(x, ξ), and inspired by the previous procedure we will prove that the energy density of the ogenized functional can be represented by means of a cell-problem formula (see Proposition???) obtained by an approximation technique. This is possible thanks to the fact that the power-law approximation described above holds true for very general f (see [?]). As a consequence we obtain that the ogenization and the power-law approximation commute as summarized by the following diagramm F p,ε Γ(L ) p F ε Γ(L p ) ε 0 ε 0 Γ(L ) F p Γ(L ) p F the down arrow being proved in Proposition?? and the left arrow beung proved in Theorem??. Asecond important step in our proof of is the key remark that there is a strict relation between the class of supremal functional and a class of very degenerate functional of the form 0 if Du(x) C(x) a.e. in G(u) = 1 C(x) (Du) dx = (1.8) + otherwise, where C(x) is a convex set (see Propositions 2.1 and 2.6 in [?]). In particular the knowledge of the ogenized functional for the latter permits, with a suitable choice of the set C(x), to deduce the Γ- limsup inequality. The ogenization for functionals of the form (1.8) can be obtained as a particular case of the results by Carbone et al. (see []) for unbounded integral functionals. Since these results require convex analysis techniques, their use will force us to restrict our study to the class of convex subsets of R N. The plan of the paper is the following. In Section 2 we recall the main tools we use for our result. Section 3 will be devoted to the derivation, by approximation, of the cell problem (see Proposition 2.1). In Section 3 we state and prove the ogenization Theorem (Theorem 4.1). Finally in Section 4 we comment the result and we show that under some additional assumptions on the function f it is possible to prove the ogenization result for a larger class of sets. 2 Formulation of the problem and preliminaries The aim of this paper is to give an ogenization result for a sequence of supremal functionals in W 1, (), i.e. functionals of the form F ε (u) = ess sup f( x,du(x)), (2.9) x ε where f(x, ξ) is a Borel function and 1-periodic in the second variable. Anecessary and sufficient condition for the lower semicontinuity in L for functionals of this type has been proved in [?] and is given by the following two conditions: 3

4 (i) (lower semicontinuity) f(x, ) is lower semicontinuous for a.e. (ii) (level convexity) f(x, ) is level convex for a.e. x, i.e. for every t R the level set ξ R N : f(x, ξ) t is convex. Remark that the level convexity can be equivalently stated as follows: for any λ (0, 1) and for all ξ 1,ξ 2 R N, then f (λξ 1 +(1 λ)ξ 2 ) f (ξ 1 ) f (ξ 2 ), for a.e. x. In addition we require for f a growth contion from below: (iii) (coercivity) there exists a positive constant c such that f(x, ξ) c ξ for a.e. x R N In order to study the asymptotic behaviour of F ε as ε 0 we will use the notion of Γ-convergence introduced by De Giorgi. For convenience of the reader let us recall the definition (more details on Γ-convergence and ogenization theory can be found for instance in [?], [?] and [?]). We say that a given sequence of functionals G ε defined in a metric space X, Γ-converges to the functional G, asε 0, if the following properties hold (a) For every u X and for every sequence u ε converging to u in X we have G(u) lim inf G ε(u ε ). (b) For every u X there exists a sequence u ε (recovering sequence) such that G(u) lim sup G ε (u ε ). We will refer to (a) as the Γ(X)-liminf inequality and to (b) as the Γ(X)-limsup inequality. former is actually equivalent to The G(u) inflim inf G ε(u ε ): u ε u in X := Γ(X)- lim inf G ε (u), while the latter gives G(u) inflim sup G ε (u ε ): u ε u in X := Γ(X)- lim sup G ε (u). We will use the notion of Γ convergence both for the ogenization in L p (), for 1 <p + and for the power-law approximation, as p +. In the latter the definition above will be applied with ɛ =. Akey step in order to use an approximation argument is the following result proved in [?]. Theorem 2.1 Let R N be a bounded open set and let f : R N [0, + ] satisfyingconditions (i) (iii). For any p>n, we define the functional ( F p (u) := f p (x, Du(x))dx if u W 1,p ((0, 1) N ) (0,1) N + otherwise. Then, the family (F p ) p>n Γ-converges (as p goes to + ) tof : C() [0, + ] (??) defined by ess sup f(x, Du(x)) if u W 1, ((0, 1) N ) F(u) := x + otherwise, with respect to the topology of the uniform convergence. Remark 2.2 In this kind of approximation results the fact that the sequence F p actually converges pointwise to the functional F, implies that the recovery sequence can be always choose to be a constant sequence. This guarantees that the sequence F p converges also if one impose any kind of boundary condition. 4

5 3 Power-law approximation and cell-problem In order to study the Γ-limit for F ε and give an explicite representation of it, we will follow the strategy of [?]. In view of the approximation result given by Theorem 2.1 let us consider the functional ( F p,ε (u, ) := f p ( x ε,du(x))dx (3.10) For any fixed 1 <p<+ one can consider the Γ-limit as ε 0, i.e. the ogenized functional Fp of F p,ε. If we make the extra assumption that an estimate from above for the function f(x, ξ) holds, e.g. f(x, ξ) C(1 + ξ ), then the function f p (x, ξ) satisfies the standard growth condition which permits to characterized the ogenizetion of F p,ε as ε 0. Namely for any bounded open set the sequence F p,ε (u, ) Γ(L p )-converges to the functional ( Fp (u, ) := fp (Du(x))dx) (3.11) where the energy density fp is given by the following cell-problem formula fp (ξ) :=(f p ) (ξ) = inf f p (x, ξ + Du(x))dx : u W 1,p # ((0, 1)N ) (3.12) (the space W 1,p # ((0, 1)N ) being defined by W 1,p # ((0, 1)N ):=u W 1,p loc (RN ):u is 1-periodic for all 1 <p + ). However in the general case, when no estimates from above are assumed, this Γ-convergence result and the corresponding cell-problem formula are still true under additional assumptions on and they have been proved by Carbone et al. in [?]. Their technique is based on convex analysis arguments and thus it requires the assumption that is a convex open subset of R N. For our ogenization result we also assume convex. In fact, even if, adding a control from above for the function f, we could avoid the use of the result in [?] for this first step, we will need it anyway for the study of the Γ-limsup of F ε. For the convenience of the reader, in the next section we will state the ogenization result for unbounded functionals proved in [?] for the particular case of the ogenization of functionals whose energy density is the indicator of a convex set C(x) (see Theorem 4.4). Using Theorem??, from (3.12) we can derive a cell-problem formula as p + which will be our candidate for the representation of the ogenized functional of F ε, i.e. f (ξ) = inf ess sup f(x, ξ + Du(x)) : u W 1, # ((0, 1)N ). (3.13) Remark 3.1 Let us if f satisfies conditions (i) (iii) the functional ess sup f(x, ξ + Du(x)) is lower semicontinuous and coercive in W 1, # ((0, 1)N ) and thus the infimum in the definition of f (ξ) is achieved, for any fixed ξ R N. In fact, as a consequence of the Ascoli-Arzela theorem, any minimizing sequence is conpact, up to a translation, and hence up to a subsequence it must converge to a minimum point. Moreover it can be easily checked that the function f (ξ) defined by (3.13) is a level convex function. Indeed for fixed λ (0, 1) and ξ 1,ξ 2 R N there exist u 1,u 2 W 1, # ((0, 1)N ) such that f (ξ i )= ess sup x f(x, ξ i + Du i (x)) for i =1, 2 and thus f (λξ 1 +(1 λ)ξ 2 ) ess sup f(x, λ(ξ 1 + Du 1 (x))+(1 λ)(ξ 2 + Du 2 (x))) x ess sup f(x, ξ 1 + Du 1 (x)) ess sup f(x, ξ 2 + Du 2 (x)) x x = f (ξ 1 ) f (ξ 2 ) which is equivalent to the level convexity of f.. 5

6 We have the following result. Lemma 3.2 Let f : R N R N [0, + ] be a Borel function, 1-periodic in the first variable satisfying conditions (i) (iii), then 1. for every ξ R N ( lim f p (ξ) = f (ξ), (3.14) 2. for all open bounded sets R N and for all u W 1, (), ( lim fp (Du(x))dx) = ess sup f (Du(x)). (3.15) x ( (f Proof. In order to prove (3.14), let us fix ξ R N p, and observe that by definition, ) is p N a non-decreasing sequence such that ( f p (ξ) f (ξ), p 1. (3.16) ( (f p, Thus, there exists the limit of ) for p + and we have p N ( lim f p (ξ) f (ξ). (3.17) As a consequence, the first inequality is proved. In order to prove the reverse inequality, without loss of ( generality, assume that lim f p (ξ) R. Let us consider the sequence (Fp ) p>n, F p : C((0, 1) N ) [0, + ], and the functional F : C((0, 1) N ) [0, + ], respectively defined by ( F p (v) := f p (x, Dv(x)+ξ)dx if v W 1,p # ((0, 1)N ) (0,1) N + otherwise, and F(v) := ess sup f(x, Dv(x)+ξ) x if v W 1, # ((0, 1)N ) + otherwise. For fixed ε>0, for every p N there exists u p W 1,p # ((0, 1)N ) such that F p (u p ) (fp (ξ) + ε. By hypothesis 2, there exists C>0such that Du p L q C for every p>q>n.since a translation does not modify the values of the functionals, without loss of generality, we can assume that u p (x)dx =0. By Poincaré-Wirtinger inequality, we obtain that there exists C 1 > 0 such that u p W 1,q C 1 for every p>q>n.in particular, there exists u W 1, # ((0, 1)N ) such that (u p ) p converges uniformly to u,as p goes to +. By Corollary 2.1, we have that (F p ) p>n Γ- converges to F with respect to the topology of the uniform convergence. Thus, also by definition of f, it follows f (ξ) F(u ) lim inf F ( p(u p ) lim inf f p (ξ) + ε. By arbitrariness of ε, we obtain the reverse inequality and conclude the proof of (3.14). In order to prove (3.15), first of all we observe that by (3.16) we easily obtain an inequality: ( ( lim fp (Du(x))dx) lim (f ) p (Du(x))dx = ess sup f (Du(x)). x 6

7 To obtain the reverse one, fix ε>0 and u W 1, (). Define E ε := x : f (Du(x)) > ess sup f (Du(x)) ε x 2 m ε := L(E ε ) > 0 and fix 0 < δ << m ε. By (3.14) and Egorov s theorem, there exists F δ such that L(F δ ) δ and lim (fp f L (\F δ ) =0. In particular there exists p 0 = p 0 (ε) such that if p p 0 then, (f p (Du(x)) f (Du(x)) ε 2 x \ F δ. Let us define E p ε := x : ( f p (Du(x)) > ess sup f (Du) ε. x Now L(E ε \ F δ ) > 0 and if x E ε \ F δ, then for every p p 0 we have ( f p (Du(x)) ess sup f (Du(x)) ε, x i.e. for every p p 0 (E ε \ F δ ) Eε p and thus for every p p 0, L(Eε p ) > 0. Finally, ( ( fp (Du(x))dx) fp (Du(x))dx Eε p ( (ess sup f (Du(x)) ε) p dx Eε p x = L(Eε p (ess sup f (Du(x)) ε) x L(E ε \ F δ (ess sup f (Du(x)) ε) x for every p p 0. This implies ( lim fp (Du(x))dx) ess sup f (Du(x)) ε x and thus, by arbitrariness on ε, we have (3.15). We are now in a position to study the Γ-limit as p of the functionals F p defined in (3.11). Theorem 3.3 Let f : R N R N [0, + ] be a Borel function, 1-periodic in the first variable satisfying conditions (i) (iii), let be a bounded subset of R N and, for any p>n, let Fp be the functional defined by (3.11), then where f is given by (3.13). Γ(L )- lim F p (u) =F (u) := sup f (Du(x)) Proof. For every pfp is a convex and l.s.c. function such that fp (ξ) c p ξ p b p (thanks to assumption 2). Thus, for every p, the functional ( F p, (u) = fp (Du(x))dx 7

8 is lower semicontinuous in W 1,p with respect to the L p topology (see Remark 5.4 in [4]). In particular it is lower semicontinuous in W 1, with respect to the L topology. Since the sequence ( F p,) is p increasing and by Lemma 3.2 it converges pointwise to F as p, by Remark 5.5. in [8] we have that Γ(L )- lim F p, = F. Remark 3.4 The ogenization result for integral functionals together with the previous theorem show that ( ) Γ(L )- lim Γ(L p )- lim F ε,p (u) = sup f (Du) 4 The ogenization result The present section is devoted to the statement and the proof of the ogenization result. In particular we will show that the two limits in Remark 3.4 commute and the functional F can be obtained directly taking the limit as ε 0ofF ε. Theorem 4.1 Let f : R N R N [0, + ] be a Borel function, 1-periodic in the first variable and satisfyingconditions (i) (iii). Then for any convex bounded open subset of R N, the sequence F ε : W 1, () [0, + ], ( x ) F ε (u) = ess sup f x ε,du(x), Γ(L )-converges to the functional F : W 1, () [0, + ] defined by F (u) := ess sup f (Du(x)), (4.18) x where f : R N [0, + ] is a level convex function given by the cell-problem formula f (ξ) := inf ess sup f(x, ξ + Du(x)) : u W 1, # ((0, 1)N ) x (4.19) for all ξ R N We will prove Theorem 4.1 in two steps: the Γ-liminf inequality (Proposition 4.2) will be obtained as a consequence of the L p approximation of the supremal functional and ogenization result for integral functionals; for the Γ-limsup inequality (Proposition 4.5) we will use an ogenization result for unbounded integral functionals (see [?]) which we will state in a convenient form in Theorem 4.4. In this second step we will need the convexity of the set. Proposition 4.2 Let f : R N R N [0, + ] be a Borel function, 1-periodic in the first variable and satisfyingconditions (i) (iii). Then for any open bounded set R N and for all u W 1, () we have Γ(L )- lim inf F ε(u) F (u). Proof. It is sufficient to prove that for every (u ε ) ε,u W 1, (), such that u ε u in L () as ε 0 we have: ( f (Du(x))dx) p, lim inf F ε(u ε )L( (4.20) for each p 1. Indeed, letting p + in (4.20) and applying (3.15) we can conclude that ess sup f (Du(x)) lim inf F ε(u ε ), x 8

9 which gives us the thesis. In order to prove (4.20), fix u W 1, () and u ε u in L () as ε 0. Let us observe that, by definition, ( lim inf f p ( x ε(x))dx) ε,du lim inf F ε(u ε, )L(. So, if f p was a convex function satisfying a growth condition of order p, we could apply Theorem?? and easily obtain (4.20). Since f is only a level convex function satisfying a coercivity condition, we need to introduce an auxiliary sequence of functions φ p,k which fulfill the hypotheses of Theorem?? and such that φ p,k approximate fp. Let us define (f p ) (x, ) as the greatest convex function minorant f p (x, ) and observe that f p = min (f p ) (x, ξ + Du(x))dx : u W 1,p # ((0, 1)N ). (4.21) Moreover, for fixed k N, let us define f p,k (x, ξ) :=min(f p ) (x, ξ),k(1 + ξ p ) and φ p,k (x, ξ) := (f p,k ) (x, ξ). By hypothesis 2 for every k c p,wehave c p ξ p φ p,k (x, ξ) k(1 + ξ p ). (4.22) Thus, for every k N and p 1, such that k c p the function φ p,k satisfies the hypotheses of Theorem??. Moreover since u ε u in L () implies u ε u in L p () we obtain that, ( ( φ p,k (Du(x))dx) lim inf lim inf ( φ p,k ( x ε(x))dx) ε,du f p ( x ε(x))dx) ε,du lim inf F ε(u ε, )L( (4.23) where φ p,k (ξ) := inf φ p,k (x, ξ + Du(x))dx : u W 1,p # ((0, 1)N ). Observe now that the sequence (φ p,k ) k is increasing. So if we prove that sup φ p,k (ξ) =fp (ξ) (4.24) k for all ξ R N, then, passing to the limit with respect to k in (4.23) and applying the Monotone Convergence Theorem, we obtain (4.20). Thus, it remains only to prove (4.24). First, we observe that by construction and by (4.21), we get the first inequality sup φ p,k (ξ) fp (ξ) for all ξ R N. k In order to prove the reverse inequality we first show that sup φ p,k (x, ξ) =(f p ) (x, ξ) (4.25) k for a.e. x and for every ξ R N. Indeed, let H such that L(H) = 0, for every x \ H, (f p ) (x, ) is a convex function. Fix ξ o R N and β<(f p ) (x, ξ o ), by definition of (f p ) there exist a R N,b Rsuch that a ξ + b (f p ) (x, ξ) for every ξ R N, β<a ξ 0 + b (f p ) (4.26) (x, ξ 0 ). Let k N such that a ξ+b k( ξ p +1) for every ξ R N. By (4.26), a ξ+b f p,k (x, ξ) for every ξ R N which implies a ξ + b φ p,k (x, ξ) for every ξ R N. Moreover β<a ξ 0 + b φ p,k (x, ξ 0 ) (f p ) (x, ξ 0 ) 9

10 and this concludes the proof of (4.25). Without loss of generality we can assume that sup k φ p,k (ξ) R. For fixed ε>0, for every k N there exists u k W 1,p # ((0, 1)N ) such that φ p,k (ξ)+ε φ p,k (x, ξ + Du k (x))dx. By (4.22), Du k L p C 1 and since we can always assume that u k (x)dx = 0, by Poincaré inequality we obtain that u k W 1,p ( ) C 2. So there exist a subsequence (which we denote again (u k ) k ) and v W 1,p # ((0, 1)N ) such that (u k ) k converges weakly to v in W 1,p ((0, 1) N ). Since the sequence (φ p,k ) k is increasing and φ p,k0 (x, ) is convex for every k 0,wehave sup φ p,k (ξ)+ε lim inf φ p,k0 (x, ξ + Du k (x))dx k k φ p,k0 (x, ξ + Dv(x))dx for every k 0 N. By (4.25), by the Monotone Convergence Theorem and by (4.21), we obtain sup φ p,k (ξ)+ε sup φ p,k (x, ξ + Dv(x))dx k k = (f p ) (x, ξ + Dv(x))dx fp (ξ). This concludes the proof of (4.24).????? Remark 4.3 Under the hypotheses of Lemma 3.2, following the proof of Proposition 4.2 we have that for every (u ε ) ε,u W 1,p (), such that u ε u in L p () as ε 0wehave: ( fp (Du(x))dx) lim inf ( (f p ) ( x ε,du(x))dx for each p N. Moreover, if is a convex open set, by proceeding as in [6], we can prove the existence of a recovering sequence for the Γ-limsup and obtain that Γ(L p ) lim( (f p ) ( x ( ε,du(x))dx = fp (Du(x))dx). Thanks to Proposition 1.16 and Example 3.11 in [8], we can conclude that Γ(L p ) lim( f p ( x ( ε,du(x))dx = fp (Du(x))dx). The proof of the Γ-limsup inequality follows from a repeated application of the ogenization result for unbounded convex integral functionals due to Carbone et al. ([?]), which we state in the following particular case. Theorem 4.4 (Theorem 3.10, [?]) Let be a convex bounded subset of R N and let C(x) a 1-periodic set function such that C(x) is a convex set for any x and there exists R>0 such that C(x) B R (0) a.e. x. (4.27) 10

11 Let G ε : W 1, () [0, + ] be defined by G ε (u) := 1 C( x ε ) (Du(x))dx, then G ε Γ(L )-converges as ε goes to zero to the ogeneous functional G (u) := g (Du(x)), where, for ξ R N, g (ξ) := inf 1 C(x) (ξ + Du(x))dx : u W 1,p # ((0, 1)N ). (4.28) In the following Proposition we prove the Γ-limsup inequality. Proposition 4.5 Let f : R N R N [0, + ] be a Borel function, 1-periodic in the first variable and satisfyingconditions (i) (iii). Then for any convex open bounded set R N and for all u W 1, (), we have Γ(L )- lim sup F ε (u, ) f (u, ). Proof. Fix ū W 1, () and let M = ess sup f (Dū). Our aim is to find a sequence (u ε ) ε such that x F ε (u ε ) M. u ε ū in L () as ε 0, and lim sup For every x let us define C(x) :=ξ R : f(x, ξ) M and C := ξ R : f (ξ) M. Since f(x, ) is level convex, the set C(x) is convex for a.e. x and, thanks to condition (iii) (the coercivity of f) it satisfies the property (4.27) of Theorem 4.4. Thus we have that Γ(L ) lim G ε (u, ) = g (Du(x)), (4.29) where, for ξ R N, g (ξ) is defined by (4.28). Let us observe that g (ξ) =1 C (ξ) for all ξ R N. By definition of g, it is sufficient to prove that g (ξ) =0 1 C (ξ) =0. Now we have that g (ξ) =0 u W 1, # ((0, 1)N ) such that 1 C(x) (ξ + Du(x))dx =0, u W 1, # ((0, 1)N ) such that f(x, ξ + Du(x)) M for a.e. x, u W 1, # ((0, 1)N ) such that ess sup f(x, ξ + Du(x)) M. By Rem 3.1, the last holds if and only if there exists u W 1, # ((0, 1)N ) such that f (ξ) M, i.e. if and only if 1 C (ξ) =0. Using this and (4.29) we have that there exists a sequence (u ε ) ε W 1, () such that u ε ū in L (), as ε 0, and lim sup G ε (u ε, ) 1 C (Dū(x))dx =0. In particular, there exists ε 0 > 0 such that, for every ε 0, G ε (u ε, ) = 0, i.e. 1 C ( x ε ) (Du ε(x))=0 x for a.e. x, i. e. ess sup f( x x ε,du ε(x)) M which implies F ε (u ε, ) M. Summarizing, we have F ε (u ε, ) M for every ε 0, and this concludes the proof. 11

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