Homogenization of non-uniformly bounded periodic diffusion energies in dimension two

Homogenization of non-uniformly bounded periodic diffusion energies in dimension two Andrea BRAIDES∗Marc BRIANE†Juan CASADO-D´ IAZ‡ October 21, 2008 Abstract This paper deals with the homogenization of two-dimensional oscillating convex functionals, the densities of which are equicoercive but not uniformly bounded from above. Using a uniform-convergence result for the minimizer, which holds for this type of scalar problems in dimension two, we prove in particular that the limit energy is local and recover the validity of the analog of the well-known periodic homogenization formula in this degenerate case. However, in the present context the classical argument leading to integral representation based on the use of cut-off functions is useless due to the unboundedness of the densities. In its place we build sequences with bounded energy, which converge uniformly to piecewise-affine functions, taking pointwise extrema of recovery sequences for affine functions. Keywords: Homogenization – Γ-convergence – Periodic – Diffusion – Dimension two Mathematics Subject Classification: 35B27 – 35J60 1 Introduction General homogenization theorems ensure that the limit of oscillating functionals of the form ZΩ fnx εn,∇udx with domain some W1,p Sobolev space is a homogeneous integral of the same form ZΩ fhom(∇u)dx provided the function fis periodic in the first variable and satisfies the ‘standard p-growth conditions’ c1|ξ|p−1≤f(y, ξ)≤c2(1 + |ξ|p) (see, e.g., [4]). This result, up to the use of asymptotic homogenization formulas to describe fhom in the vector case, is valid in any dimension and its proof is usually achieved using a technical argument due to De Giorgi, which consists in the use of ‘cut-off’ functions ϕnin the construction of recovery sequences of the form vnϕn+ (1 −ϕn)unas a convex combination of two recovery sequences. The use of the p-growth condition allows to optimize the choice of these ϕn. This argument ∗Dipartimento di Matematica, Universit`a Roma ‘Tor Vergata’, [email protected] †Centre de Math´ematiques I.N.S.A. de Rennes & I.R.M.A.R., m[email protected] ‡Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, jcasado[email protected] 1 is used to ‘glue’ optimal sequences on overlapping sets, match boundary conditions, etc., and is stable under small variations of funder the above-mentioned growth conditions (see [4]). For functionals not uniformly satisfying a p-growth condition, this result fails. In particular the limit of energies of the form Fn(u) = ZΩ fnx εn,∇udx, where fnare periodic in the first variable and satisfy ‘degenerate standard p-growth conditions’ cn 1|ξ|p−1≤f(y, ξ)≤cn 2(1 + |ξ|p) with cn 1possibly vanishing and cn 2possibly diverging, a ‘local’ representation of the limit energy through the single variable umay fail. For quadratic energies it can be represented as a Dirichlet form (see [17]), or as a multi-phase energy (see [1], [6], [8], [9], [13], [15], [16]). Results by Camar-Eddine and Seppecher [10] determine that a wide class of local and non-local quadratic forms can be reached as Γ-limit of usual local Dirichlet-type integrals with degenerate coefficients. The object of this paper is the homogenization of (nonlinear) integral functionals Fn as above, where Ω is a bounded open set of R2and uis scalar, when fnsatisfies very mild growth conditions from above (see (2.1)–(2.3) below). In the simplest (linear and isotropic) case this can be translated into the Γ-convergence of oscillating functionals of the form Fn(u) = ZΩ anx εn|∇u|2dx, where an≥1 are 1-periodic but anare not bounded in L∞. In this case many of the usual techniques of Γ-convergence hinted at above do not work as they are usually stated, but must be carefully modified. This can be seen by examining a sequence wn:= ϕnun+ (1 −ϕn)vnobtained by “joining” two sequences unand vnwith bounded energy. Its energy can be estimated by the energies along the sequences unand vn, and a term depending on ∇ϕnand un−vn. In the linear case above this remainder term takes the form ZΩ anx εn|∇ϕn|2|un−vn|2dx, and can be made arbitrarily small when un−vntends to zero in L2, upon suitably choosing ϕn, if anis bounded in L∞. For unbounded coefficients, for such an argument to work some stronger convergence is required. In the two-dimensional case the compactness result of Briane and Casado-Diaz [7] ensures that we can restrict to sequences such that un−vn converges to zero uniformly, so that the error above is estimated by k∇ϕnk2 ∞kun−vnk2 ∞ZΩ anx εndx ≤ |Ω| k∇ϕnk2 ∞sup n kankL1((0,1)2)kun−vnk2 ∞, which shows that the L1-boundedness of ancan be used in the cut-off argument. In place of an L1-boundedness assumption we will suppose that lim n→∞ fhom n(ξ)≤¯ b(1 + |ξ|p) for all ξ∈R2, where the energy density fhom nis given by the cell-problem formula (2.4). This assumption clearly holds if fnsatisfies an L1-boundedness hypothesis of the type fn(y, ξ)≤bn(y) (1 + |ξ|p), with supnkbnkL1((0,1)2)<∞, but is more general and covers the case of domains with strong inclusions. 2 Under such a general assumption we bypass the cut-off arguments above, using the specificity of the scalar setting coupled with the improved convergence of recovery sequences. To exemplify our approach, we can consider the simplest case of the construction of optimal sequences for a function of the form u=u1∨u2(∨denotes the maximum) with uiaffine. If ui nare optimal sequences for uithen we can simply set un:= u1 n∨u2 n. The uniform convergence of ui nallows then to estimate the error in terms of the size of a small neighbourhood of the set {u1=u2}. A technical argument allows then to carry on this construction to optimal sequences for arbitrary piecewise-affine functions and then by density to the whole space W1,p. This proves one of the two inequalities – namely, the Γ-limsup inequality – of Γ-convergence. To prove the Γ-liminf inequality we have found it convenient to use the Fonseca-M¨uller blow-up technique, which allows to reduce to the study of converging sequences when the target function is linear ξ·x. A similar argument as above allows then to modify such sequences so that it satisfies periodic boundary conditions, which allows an estimate with the energy densities fhom n(ξ). Again the scalar nature of the problem is heavily exploited both in the modification leading to periodic boundary conditions and in the reduction to a single cell-problem formula. The paper is organized as follows. In Section 2 we state the main result which is proved in Section 3. Section 4 is devoted to a sufficient condition permitting to derive the boundedness of fhom nin R2. Notation •for any open set ωof R2, ¯ωdenotes the closure of ωin R2; •Y:= (0,1)2; •H](Y) denotes the space of the Y-periodic functions which belong to Hloc(R2); 2 Statement of the results Let p > 1, and let Ω be a bounded open set of R2with a Lipschitz-continuous boundary. We consider a sequence of non-negative functions fn:R2×R2→[0,∞), for n≥1, satisfying the following properties: fn(·, ξ) is a Y-periodic measurable function for any ξ∈R2,(2.1) fn(y, ·) is convex with fn(y, ·)≥fn(y, 0) for a.e. y∈R2,(2.2) there exists a non-negative sequence bnsuch that |ξ|p−1≤fn(y, ξ)≤bn(1 + |ξ|p),∀ξ∈R2,a.e. y∈R2,(2.3) Remark 2.1. In (2.2) we can replace the convexity assumption by a continuity assumption. To this end, it is enough to replace the density fn(y, ·) by its convexification, which leads us to the same convergence result (see Theorem 2.3). We define, for each fixed n≥1, the “homogenized” density fhom nby the classical minimization formula (see, e.g., Chapter 14 of [4]): fhom n(ξ) := inf ZY fn(y, ξ +∇ϕ)dy :ϕ∈W1,p ](Y),for ξ∈R2.(2.4) 3 Thanks to the convexity and the bounds (2.3) satisfied by the function fn, the infimum in problem (2.4) is attained, i.e. ∀ξ∈R2,∃ϕξ n∈W1,p ](Y) such that fhom n(ξ) = ZY fny, ξ +∇ϕξ ndy. (2.5) We will use the De Giorgi Γ-convergence theory. We refer to [11], [2] or [4] for a general presentation and the basic properties of Γ-convergence. Here, we simply recall the following definition: Definition 2.2. A sequence of functionals Fn:Lp(Ω) →[0,∞] is said to Γ-converge to F:Lp(Ω) →[0,∞] for the strong topology of Lp(Ω) if, for any uin Lp(Ω), (i) the Γ-liminf inequality holds ∀un−→ ustrongly in Lp(Ω), F(u)≤lim inf n→∞ Fn(un),(2.6) (ii) the Γ-limsup inequality holds ∃¯un−→ ustrongly in Lp(Ω), F(u) = lim n→∞ Fn(¯un).(2.7) Any sequence satisfing (2.7) will be called a recovery sequence for Fn, of limit u. Let εnbe a sequence of positive numbers, which converges to 0 as n→ ∞. For any n≥1, we define the functional Fn:Lp(Ω) →[0,∞] by Fn(u):=     ZΩ fnx εn,∇udx if u∈W1,p(Ω) ∞elsewhere. (2.8) The main result of the paper is the following theorem: Theorem 2.3. Let Ωbe a bounded open set of R2, with a Lipschitz continuous boundary. In addition to conditions (2.1)–(2.3), assume that there exist a positive constant ¯ band a function fhom ∞:R2→[0,∞), such that ∀ξ∈R2,lim n→∞ fhom n(ξ) = fhom ∞(ξ)≤¯ b(1 + |ξ|p).(2.9) Then, the sequence of functionals Fndefined by (2.8) Γ-converges for the strong topology of Lp(Ω), to the functional F∞defined by F∞(u):=ZΩ fhom ∞(∇u)dx (2.10) for all u∈W1,p(Ω). Remark 2.4. Theorem 2.3 provides an extension of the periodic homogenization of energies even in the case of a single function; i.e., when the density fn(y, ξ) = f(y, ξ) does not depend on nand satisfies the growth condition |ξ|p−1≤f(y, ξ)≤b(y) (1 + |ξ|p),∀ξ∈R2,a.e. y∈R2, where b∈L1 ](Y). The classical framework of the periodic homogenization is based on the stronger assumption b∈L∞ ](Y), but holds true in any dimension and for non-convex vector-valued problems (see, e.g., Section 21.3 of [4]). The two-dimensional setting allows us to relax the right-hand side of the growth estimate (2.3), with a sequence bnwhich is not necessarily bounded in L1 ](Y). As a consequence we need to modify the definitions (2.8) of Fn and (2.4) of fhom nby assuming the continuity of the functions. 4 Remark 2.5. We can replace the assumption that 0 is an absolute minimizer of fn(y, ·) for a.e. y∈R2, by the following more general one: There exist a function θ: [0,∞)→[0,∞) and a sequence of functions ϕnin C](εnY)∩ W1,p ](εnY), such that for any n≥1, lim t→0θ(t) = 0,∀x1, x2∈R2,|ϕn(x1)−ϕn(x2)| ≤ θ(|x1−x2|),(2.11) ∇ϕn(εny) is an absolute minimizer of fn(y, ·) for a.e. y∈R2.(2.12) For example, the sequence defined by ϕn(x):=εnϕ(x εn), for x∈R2, where ϕ∈W1,∞ ](Y), satisfies condition (2.11) with θ(t):=k∇ϕk∞t. 3 Proof of the results 3.1 A uniform-convergence result We have the following result which extends the uniform convergence result obtained in the linear framework of [7]: Proposition 3.1. Let Ωbe a bounded open set of R2, with a Lipschitz continuous boundary. Let fn:R2×R→[0,∞)be functions satisfying conditions (2.1), (2.3) and (2.12). Consider a function u∈W1,p(Ω) ∩C(¯ Ω), and a sequence ˆunin W1,p(Ω) which strongly converges to uin Lp(Ω), with ZΩ fnx εn,∇ˆundx ≤c. (3.1) Let Ω0be an open subset of Ω. Then, there exist a subsequence of n, still denoted by n, and a sequence unin W1,p(Ω) which satisfies the convergences un−* u weakly in W1,p(Ω) and un−→ ustrongly in L∞ loc(Ω0),(3.2) and the energy estimate ZΩ0 fnx εn,∇undx ≤ZΩ0 fnx εn,∇ˆundx +o(1).(3.3) Moreover, for any open subsets ω, ˜ωof Ω, with ¯ω⊂˜ω, the sequence unsatisfies Zω fnx εn,∇undx ≤Z˜ω fnx εn,∇ˆundx +o(1).(3.4) Remark 3.2. In Proposition 3.1 the case p∈(1,2] is the most relevant, since in dimension two the embedding of W1,p(Ω) in C(¯ Ω) is compact for p > 2. The result of Proposition 3.1 also extends to the following periodic case with the sequence of functionals F],ξ n, for ξ∈R2, defined by F],ξ n(ϕ):=ZY fnnx, ∇ϕ(x)dx, for ϕ∈W1,p ](Y).(3.5) 5 Proposition 3.3. For n≥1and ξ∈R2, consider ϕξ n∈W1,p ](Y)satisfying (2.5). Then, there exists a sequence ψnwhich converges to zero weakly in W1,p ](Y)and strongly in L∞ ](Y), such that ZY fnnx, ξ +∇ψn(x)dx =ZY fnnx, ξ +∇ϕξ n(nx)dx +o(1) = fhom n(ξ) + o(1).(3.6) Moreover, for any regular bounded open sets ω, ˜ωof R2, with ¯ω⊂˜ω, we have Zω fnnx, ξ +∇ψn(x)dx ≤ |˜ω|fhom n(ξ) + o(1).(3.7) Proposition 3.1 is based on the following maximum principle result: Lemma 3.4. Let Obe a bounded open subset of R2. Let ϕbe a function in W1,p(O) satisfying (2.11). Let g:O×R2→Rbe a function such that (i)g(·, ξ)is measurable for any ξ∈R2, (ii)g(x, ·)is strictly convex for a.e. x∈O, (iii)gsatisfies the growth condition |ξ|p−1≤g(x, ξ)≤β(x) (1 + |ξ|p),∀ξ∈R2,a.e. x∈O, where β∈L1(O), (iv)∇ϕ(x)is an absolute minimizer of g(x, ·)for a.e. x∈O. Let G:W1,p(O)→[0,∞]be the functional defined by G(u):=ZO g(x, ∇u)dx, for u∈W1,p(O). For ˆu∈W1,p(O)∩C(¯ O)with G(ˆu)<∞, consider the function u∈W1,p(O)defined by the minimization problem G(u) = min nG(v) : v−ˆu∈W1,p 0(O)o<∞. Then, we have the following maximum principle min ∂O (ˆu−ϕ)≤u−ϕ≤max ∂O (ˆu−ϕ)a.e. in O. Proof of Proposition 3.1. The proof is an adaptation of the proof of Theorem 2.1 in [7] to the present nonlinear framework. Therefore, we will give the main steps of the proof without specifying the details. Define the function gn: Ω ×R2→[0,∞) by gn(x, ξ):=fnx εn, ξ+1 n|ξ− ∇ϕn(x)|p,for (x, ξ)∈Ω×R2, and the functional Gn:W1,p(Ω) →[0,∞] by Gn(u):=ZΩ0 gn(x, ∇u)dx, for u∈W1,p(Ω). Note that, by the convexity of fn(y, ·) and (2.12), the function gn(x, ·) is a strictly convex function in R2with ∇ϕn(x) as an absolute minimum. 6 Using a density argument and the continuity of the functional v7→ RΩ0fn(x, ∇v)dx in W1,p(Ω), we can assume that ˆunis regular without modifying the right-hand side of (3.3). By estimate (3.1) combined with the equicoercivity of gn(x, ·) (as a consequence of (2.3)) the sequence ˆunis bounded in W1,p(Ω) and thus weakly converges to uin W1,p(Ω). Then, by virtue of the regularity of Ω, up to a subsequence, ˆunconverges uniformly to uin a relatively closed subset Kof Ω, such that for a given q∈(1, p), the q-capacity Cq(Ω \K) of Ω \Kcan be chosen arbitrarily small. By Lemma 2.8 of [7] (which is specific to dimension two) the diameter of any connected component Oof Ω \Kis bounded by a constant times Cq(Ω \K) 1 2−q. Therefore, there exists an increasing sequence nk,k≥1, of positive integers and a sequence Kkof relatively closed subsets of Ω such that ∀n≥nk,kˆun−ukL∞(Kk)≤1 k,(3.8) and for any connected component Oof Ω \Kk, diam (O)≤1 k.(3.9) Now, for any n∈[nk, nk+1), define the function un∈W1,p(Ω) by the following procedure: •in any connected component Oof Ω \Kksuch that O⊂Ω0,unis defined by the minimization problem ZO gn(x, ∇un)dx = min ZO gn(x, ∇v)dx :v−ˆun∈W1,p 0(O),(3.10) •un:= ˆunelsewhere. Taking into account (3.1) it is easy to check that un∈W1,p(Ω) and un−ˆun∈W1,p 0(Ω). Thanks to Lemma 3.4 we have, for any connected component of Ω \Kk, ∀n∈[nk, nk+1),min ∂O (ˆun−ϕn)≤un−ϕn≤max ∂O (ˆun−ϕn) a.e. in O. (3.11) Consider the increasing sequence of open subsets of Ω0defined by Ω0 k:= x∈Ω0: dist (x, ∂Ω0)>2 k,for k≥1. Note that by estimate (3.9) any connected component Osuch that O∩Ω0 k6= Ø, satisfies O∩∂Ω = Ø and thus ∂O ⊂Kk. Then, estimates (3.8), (3.11) and the triangle inequality imply that ∀n≥nk,kun−ukL∞(Ω0 k)≤1 k+ sup x,y ∈Ω |x−y|≤ 1 k |u(x)−u(y)|+|ϕn(x)−ϕn(y)|. This, combined with the uniform continuity of uin ¯ Ω and (2.11), yields lim k→∞ sup n≥nk kun−ukL∞(Ω0 k)= 0, which implies the uniform convergence (3.2). On the other hand, by the construction of ˆunwe have ∀n≥1, un−ˆun∈W1,p 0(Ω0) and Gn(un) = ZΩ0 gn(x, ∇un)dx ≤Gn(ˆun).(3.12) 7 Estimate (3.12) combined with the equicoercivity of gn(x, ·), estimate (3.1) and the boundedness of ˆunin W1,p(Ω), implies that unis also bounded in W1,p(Ω). Therefore, unsatisfies the weak convergence in (3.2). Again by (3.12) we get Gn(un) = ZΩ0 fnx εn,∇undx +1 nZΩ0 |∇un− ∇ϕn|pdx =ZΩ0 fnx εn,∇undx +o(1) ≤Gn(ˆun) + o(1) = ZΩ0 fnx εn,∇ˆundx +o(1), which yields (3.3). Finally, for klarge enough, any connected component Oof Ω \Kkwith O∩¯ω6= Ø, satisfies O⊂˜ω\Kk. Hence, from the definitions of gnand unwe deduce that for any n∈[nk, nk+1), Zω\Kk fnx εn,∇undx ≤X O⊂˜ω\KkZO fnx εn,∇undx ≤Z˜ω\Kk fnx εn,∇ˆundx +o(1). This combined with the equality un= ˆunin Kk, implies (3.4) and concludes the proof. Proof of Proposition 3.3. Let us start by the following remark: In Proposition 3.1, when Ω := (−k, k)2, for an integer k≥2, and ˆunis a sequence of Y-periodic functions which weakly converges to uin W1,p(Ω), the closed sets Kon which the convergence of ˆun is uniform are Y-periodic. Indeed, the open sets Ω \Kof arbitrary small capacity are built from sets of the type {x∈Ω : |ˆun(x)−u(x)| ≥ ε},ε > 0, (see, e.g., Theorem 7 of [12]) which are clearly Y-periodic. Therefore, the sequence undefined by (3.10) is also Y-periodic. So, the procedure of Proposition 3.1 preserves the periodicity. Let ξ∈R2. First of all, using a density argument and the continuity of the functional ϕ7→ RYfn(y, ξ+∇ϕ)dy in W1,p ](Y), there exists a sequence ˆ ψnin C1 ](Y) which is bounded in W1,p ](Y) and satisfies ZY fny, ξ +∇ˆ ψn(y)dy =ZY fny, ξ +∇ϕξ n(y)dy +o(1) = fhom n(ξ) + o(1).(3.13) On the other hand, for any integer k≥2, the sequence F],ξ ndefined by (3.5) reads as F],ξ n(ϕ) := 1 4k2Z(−k,k)2 fnnx, ξ +∇ϕ(x)dx, for ϕ∈W1,p ](Y), and the continuous functions 1 nˆ ψn(nx) weakly converge to zero (continuous) in W1,p ](Y). Then, by the preliminary remark there exists a sequence ψnwhich weakly converges to zero in W1,p ](Y) and strongly in L∞ ](Y), such that F],ξ n(ψn) = ZY fnnx, ξ +∇ψn(x)dx ≤F],ξ n1 nˆ ψn(nx)+o(1) = ZY fnnx, ξ +∇ˆ ψn(nx)dx +o(1). This, combined with (3.13) and the Y-periodicity of ˆ ψn, yields the first estimate ZY fnnx, ξ +∇ψn(x)dx ≤fhom n(ξ) + o(1).(3.14) 8 On the other hand, let ˜ ψnbe the Y-periodic function defined by ˜ ψn(y) := 1 nX κ∈{0,...,n−1}2 ψny+κ n,for y∈R2.(3.15) By the definition (2.4) of fhom n, the Y-periodicity of ˜ ψn, ψn, fn(·, ξ), and by the convexity of fn(x, ·), we have fhom n(ξ)≤ZY fny, ξ +∇˜ ψn(y)dy =ZY fnnx, ξ +∇˜ ψn(nx)dx (y=nx) ≤1 n2X κ∈{0,...,n−1}2ZY fnnx, ξ +∇ψn(x+κ n)dx =1 n2X κ∈{0,...,n−1}2Zκ n+Y fnny, ξ +∇ψn(y)dy (y=x+κ n) =ZY fnny, ξ +∇ψn(y)dy. (3.16) Therefore, (3.14) and (3.16) imply the desired estimate (3.6). On the other hand, similarly to (3.4) we obtain, owing to the construction of the function ψnfrom 1 nˆ ψn(nx), the inequality Zω fnnx, ξ +∇ψn(x)dx ≤Z˜ω fnnx, ξ +∇ˆ ψn(nx)dx +o(1). Then, by the Y-periodicity of ˆ ψncombined with the regularity of ˜ωwe get Zω fnnx, ξ +∇ψn(x)dx ≤ |˜ω|ZY fny, ξ +∇ˆ ψn(y)dy +o(1), which implies inequality (3.7) by taking into account (3.13).  Proof of Lemma 3.4. First note that the existence and the uniqueness of the function u is a consequence of the coerciveness and the strict convexity of g(x, ·) combined with G(ˆu)<∞. Set m:= min∂O(ˆu−ϕ). Since the negative part of u−ϕ−m, (u−ϕ−m)− belongs to W1,p 0(O) (see Lemma 2.7 of [7]) and ∇ϕ(x) is an absolute minimum of g(x, ·), we have G(u)≤G(u+ (u−ϕ−m)−) = Z{u−ϕ≥m} g(x, ∇u)dx +Z{u−ϕ<m} g(x, ∇ϕ)dx =ZO g(x, ∇u)dx +Z{u−ϕ<m}g(x, ∇ϕ)−g(x, ∇u)dx ≤G(u), Hence, by the convexity of Gwe deduce that G(u)≤Gu+1 2(u−ϕ−m)−≤1 2G(u) + Gu+ (u−ϕ−m)−≤G(u), which yields ZO1 2g(x, ∇u) + gx, ∇u+∇(u−ϕ−m)−−gx, ∇u+1 2∇(u−ϕ−m)−dx = 0. This combined with the strict convexity of g(x, ·) implies that ∇(u−ϕ−m)−= 0 a.e. in O. Therefore, we obtain m≤u−ϕa.e. in O. Similarly, we get u−ϕ≤max∂O(ˆu−ϕ) a.e. in O. 9 Now, consider u∈C1¯ B(ˆx, δ0)with k∇ukL∞(B(ˆx,δ0)) < ε, and define R:= (u−w0)◦z−1 which belongs to C1(O). Then, we have ∀x∈B(ˆx, δ0), u(x) = w0(x) + R(z(x)) and ∇u(x) = ∇w0(x) + Dz(x)T∇R(z(x)), which gives ∇R(z(x)) = Dz(x)T−1∇(u−w0)(x), where Tdenoted the transposition. Defining η:= −Dz(ˆx)T−1∇w0(ˆx), we get |∇R(z(x)) −η| ≤ |∇u(x)|Dz(x)−1+Dz(x)−1∇w0(x)− ∇w0(ˆx) +∇w0(ˆx)Dz(x)−1−Dz(ˆx)−1 <2εDz(ˆx)−1+ε+ε∇w0(ˆx). (4.10) On the other hand, note that η= (η1, η2) is also defined by the equality 0 = (1 −η1−η2)∇w0(ˆx) + η1∇w1(ˆx) + η2∇w2(ˆx), which by (4.8) implies that η1>0, η2>0 and η1+η2<1. Then, taking εsmall enough in (4.10) we can assume that these strict inequalities also hold for the components of ∇R(z), i.e. ∂1R(z)>0, ∂2R(z)>0 and ∂1R(z) + ∂2R(z)<1.(4.11) Now, define zn:= (w1 n−w0 n, w2 n−w0 n) and un:= w0 n+R◦znin B(ˆx, δ), with δ=δ0/2. The function unis well defined because z¯ B(ˆx, δ)is a compact subset of O, hence its distance to ∂O is positive. Since znstrongly converges to zin L∞B(ˆx, δ), we have that for nlarge enough, znB(ˆx, δ)⊂O. Clearly, unstrongly converges to uin B(ˆx, δ) and satisfies ∇un=1−∂1R(zn)−∂2R(zn)∇w0 n+∂2R(zn)∇w1 n+∂3R(zn)∇w2 n. Thanks to (4.11) and to the uniform convergence of ∂jR(zn) to ∂jR(z), we get that ∇un is a convex combination of the ∇wi n, for i= 1,2,3, hence by (4.9) we obtain that lim sup n→∞ ZB(ˆx,δ) gn(x, ∇un)dx < +∞.(4.12) Therefore, we have proved the existence of δ, ε > 0 such that for any u∈C1¯ B(ˆx, 2δ), with k∇ukL∞(B(ˆx,2δ)) < ε, there exists a sequence unin W1,p(B(ˆx, δ)) which strongly converges to uin L∞B(ˆx, δ)and satisfies (4.12). Moreover, if the support of uis contained in B(ˆx, δ), then we can easily construct a function unwith compact support in B(ˆx, δ) so that unis defined in the whole set Ω. This establishes Proposition 4.2 for any u∈C1 c(Ω) with k∇ukL∞(Ω) < ε. If udoes not satisfy this restriction, then we apply the result to v:= εu/ 2k∇ukL∞(Ω), and we consider the sequence un:= 2 k∇ukL∞(Ω) vn/ε, where vnis the sequence relating to v. We use property (4.6) to conclude. As a consequence of Proposition 4.2 we have the following result in the periodic case: Corollary 4.3. In addition to conditions (4.3)–(4.6) assume that for all ξ∈R2we have gn(x, ξ) = fn(nx, ξ)for a.e. x∈Ω, where fn(·, ξ)is Y-periodic. Also assume that there exists a non-zero function in W1,p(Ω) ∩D(G)with compact support in Ω. Then, we have C1 c(Ω) ⊂D(G). 16 Proof. Let u∈W1,p(Ω)∩D(G) be with compact support in Ω, and consider a sequence un which weakly converges to uin W1,p(Ω) and such that Gn(un) is bounded. Then, by periodicity and by a translation argument, we have that for any τ∈R2, with small enough norm, there exist a sequence uτ nin W1,p(Ω) which weakly converges to u(·+τ) in W1,p(Ω), such that (see, e.g., Chapters 23-24 of [11] for more details) lim sup n→∞ Gn(uτ n) = lim sup n→∞ Gn(un). Hence, we deduce that for any nonnegative ρ∈C∞ c(R2) and any τ1, . . . , τm∈R2, with Pm i=1 ρ(τi)>0, the function m X i=1 ρ(τi)u(·+τi) m X i=1 ρ(τi) also belongs to D(G), as well as the function x7−→ ZR2 u(x−y)ρ(y)dy ZR2 ρ(y)dy . Therefore, we are led to the case where uis a non-zero function in C∞ c(Ω) ∩D(G). Now, from Lemma 4.4 below we deduce that for any ξ∈R2, with small enough norm, there exists x∈Ω such that ∇u(x) = ξ. Using the translated functions u(·+τ) as before, we thus get that any point of Ω satisfies the assumptions of Proposition 4.2, which implies that C1 c(Ω) ⊂D(G). Lemma 4.4. Let Ωa bounded open set of Ω⊂R2. Consider a function u∈C1(Ω)∩C(¯ Ω) with u= 0 on ∂Ω, such that there exists x0∈Ωwith u(x0)6= 0. Then, for any ξ∈R2 with |ξ|<|u(x0)| max x∈∂Ω|x0−x|,(4.13) there exists x∈Ωsuch that ∇u(x) = ξ. Proof. We can assume that x0= 0 and u(0) >0. For ξ∈RN, we consider y∈¯ Ω such that u(x)−ξ·x= max y∈¯ Ωu(y)−ξ·y. If x∈∂Ω, then we have u(x) = 0 and u(0) ≤ − ξ·x≤ |ξ|max y∈∂Ω|y|, hence |ξ| ≥ u(0) maxy∈∂Ω|y|. Conversely, if |ξ|<u(0) max y∈∂Ω|y|, then xis a maximizer of y7→ u(y)−ξ·yin Ω, which implies that ∇u(x) = ξ. 17 4.3 Proof of Theorem 4.1 We need the following result which is essentially based on the continuity assumption (4.2): Lemma 4.5. Assume that the continuity condition (4.2) holds. Then, for any ξ∈R2, the sequence of functions wξ ndefined by wξ n(x):=ξ·x+1 nϕξ n(nx),x∈R2, strongly converges to ξ·xin L∞ loc(R2). Proof. Let Ω be a bounded open set of R2. The sequence wξ nclearly converges to the continuous function ξ·xweakly in W1,p(Ω). Moreover, since ϕξ nis a Y-periodic minimizer of (2.5), we have for any open set O⊂Ω, ZO fnnx, ∇wξ ndx = min ZO fn(nx, ∇wξ n+∇ϕ)dx :ϕ∈W1,p 0(O).(4.14) Then, taking into account the continuity of wξ n, the construction of the proof of Proposition 3.1 (compare (3.10) to (4.14)) shows that the sequence wξ nstrongly converges to ξ·x in L∞ loc(Ω). As a consequence of Corollary 4.3 we have that C1 c(Ω) ⊂D(F) for any bounded open set of R2. Let Ω be the unit disk of R2, and fix δ > 0. Let φ∈C1 c(1 + 2δ)Ωwith φ= 1 in (1+δ)Ω. Then, by Corollary 4.3 and Proposition 3.1 applied to the open set (1+2δ)Ω, there exists a sequence ζnwhich converges to φ(x)ξ·xweakly in W1,p(1 + 2δ)Ωand strongly in L∞(1 + δ)Ω, such that lim sup n→∞ Z(1+δ)Ω fn(nx, ∇ζn)dx < ∞.(4.15) Similarly, for a function ϕ∈C1 c(1 + δ)Ωwith 0 ≤ϕ≤1 in (1 + δ)Ω and ϕ= 1 in Ω, there exists a sequence ϕnwhich converges to ϕweakly in W1,p(1 + 2δ)Ωand strongly in L∞(1 + δ)Ω, such that lim sup n→∞ Z(1+δ)Ω fn(nx, ∇ϕn)dx < ∞.(4.16) Using truncations we can also assume that 0 ≤ϕn≤1 in (1 + δ)Ω and ϕn= 1 in Ω. On the one hand, using successively the minimization property (4.14) of wξ nand the convexity (2.2) of fn(nx, ·), we have Z(1+δ)Ω fnnx, ∇wξ ndx ≤Z(1+δ)Ω fnnx, ∇(wξ n+ϕn(ζn−wξ n)dx =Z(1+δ)Ω fnnx, ϕn∇ζn+ (1 −ϕn)∇wξ n+ (ζn−wξ n)∇ϕndx ≤1 2Z(1+δ)Ω ϕnfn(nx, 2∇ζn)dx +1 2Z(1+δ)Ω fnnx, 2(ζn−wξ n)∇ϕndx +1 2Z(1+δ)Ω (1 −ϕn)fnnx, 2∇wξ ndx, hence by estimate (4.1) we get Z(1+δ)Ω fnnx, ∇wξ ndx ≤C 2Z(1+δ)Ω fn(nx, ∇ζn)dx +C 2 ζn−wξ n  p L∞((1+δ)Ω) Z(1+δ)Ω fn(nx, ∇ϕn)dx +C 2Z(1+δ)Ω\Ω fnnx, ∇wξ ndx (since ϕn= 1 in Ω). (4.17) 18 On the other hand, the Y-periodicity of ∇wξ nimplies that Z(1+δ)Ω\Ω fnnx, ∇wξ ndx ≈ n→∞ (1 + δ)2−1 (1 + δ)2Z(1+δ)Ω fnnx, ∇wξ ndx. (4.18) Moreover, the uniform convergence of ζnand Lemma 4.5 combined with estimates (4.15) and (4.16) give C 2Z(1+δ)Ω fn(nx, ∇ζn)dx +C 2 ζn−wξ n  p L∞((1+δ)Ω) Z(1+δ)Ω fn(nx, ∇ϕn)dx ≤c. 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