3D-2D analysis of a thin film with periodic microstructure

J.-F. Babadjian and M. Baía

Abstract

The purpose of this article is to study the behavior of a heterogeneous thin film whose microstructure oscillates on a scale that is comparable to that of the thickness of the domain. The argument is based on a 3D-2D dimensional reduction through a Γ-convergence analysis, techniques of two-scale convergence and a decoupling procedure between the oscillating variable and the in-plane variable.

Keywords: dimension reduction, thin films, periodic integrands, Γ-convergence, two-scale convergence, quasiconvexity, equi-integrability.

MSC 2000 (AMS): 35E99, 35M10, 49J45, 74B20, 74K35, 74G65, 74Q05

1 Introduction and main result

In this work we study the asymptotic behavior of a heterogeneous ε-thin domain with periodic microstructure of period ε, as ε goes to zero, through a Γ-limit analysis. Techniques of two-scale convergence and a decoupling procedure between the oscillating variable and the in-plane variables are used to derive the relaxed two-dimensional energy from its three-dimensional counterpart.

Let ω be an open and bounded subset of ℝ^{2}. For each 0 < ε ≪ 1 define Ω_{ε} := ω×(−ε, ε) (Figure. 1). Consider a deformable thin body occupied by a hyperelastic material with a periodic microstructure of period ε whose reference configuration is given by the thin domain Ω_{ε}, and whose stored energy density W (ε) : Ω_{ε} × ℝ^{3×3} → ℝ is assumed to be a Carathéodory function satisfying some p-growth and coercivity conditions (1 < p < ∞). We assume that the body is clamped on the lateral boundary, that is u(x) = x on ∂ω × (−ε, ε) for all its admissible deformations, and that it is submitted to the action of regular surface traction densities g(ε) on Σ_{ε} := ω × {−ε, ε} and regular dead loads f(ε). The total energy of this body under the action of this forces is nothing but the difference between the elastic energy and the work of external forces. More precisely

E(ε)(u) := ∫_{Ω_{ε}} W (ε)(x; Du) dx − ∫_{Ω_{ε}} f(ε) · u dx − ∫_{Σ_{ε}} g(ε) · u dH^{2},

Figure 1: Thin domain Figure 2: Example of material with periodic structure of period ε in the in-plane direction (two sections)

Figure 3: Rescaled domain of unit thickness

for u ∈ V(ε) := {u ∈ W^{1,p}(Ω_{ε}; ℝ^{3}) : u(x) = x on ∂ω × (−ε, ε)}, and where H^{2} stands for the two-dimensional Hausdorff measure. It may occur that the minimization problem associated with this energy admits no solution over the set of kinematically admissible fields V(ε). However, we can introduce the notion of quasi-minimizer of E(ε), u(ε) ∈ V(ε), by requiring that

E(ε)(u(ε)) ⩽ inf_{u∈V(ε)} E(ε)(u) + ε h(ε),

where h(ε) ↘ 0^{+} when ε → 0. Note that if the minimization problem admits a solution – for instance if W (ε) is quasiconvex in its second variable – then we can take h ≡ 0.

As usual, in order to study this problem as ε → 0 we rescale the ε-thin body into a reference domain of unit thickness (see e.g. Anzellotti, Baldo and Percivale [3], Le Dret and Raoult [17], Braides, Fonseca and Francfort [10]), so that the resulting energy will be defined on a fixed body, while the dependence on ε turns out to be explicit in the transverse derivative. For this, we consider the change of variables

Ω_{ε} → Ω := ω × I, (x_{1}, x_{2}, x_{3}) ↦ (x_{1}, x_{2}, \frac{1}{ε}x_{3}) ,

and define v(x_{α}, x_{3}/ε) = u(x_{α}, x_{3}) on the rescaled cylinder Ω, where I := (−1, 1) (Figure. 3) and x_{α} := (x_{1}, x_{2}) is the in-plane variable. It is well known that the membrane theory arises at the order ε of a formal asymptotic expansion (see Fox, Raoult and Simo [16]), provided that the body forces are of order 1 and the surface loadings are of order ε. Since this energy is of order ε we divide the total energy by ε and, in addition we assume that

\{ \begin{matrix}f(ε)(x_{α}, εx_{3}) & = & f(x_{α}, x_{3}), \\ g(ε)(x_{α}, εx_{3}) & = & ε g(x_{α}, x_{3}), \end{matrix}

where f ∈ L^{p′}(Ω; ℝ^{3}), g ∈ L^{p′}(Σ; ℝ^{3}) (1/p + 1/p′ = 1) and Σ := ω × {−1, 1}. If W_{ε}(x_{α}, x_{3}; ·) = W (ε)(x_{α}, εx_{3}; ·), for fixed ε minimizing E(ε) on V(ε) is equivalent to minimizing

E_{ε}(v) := \frac{E(ε)(u)}{ε} = ∫_{Ω} W_{ε} (x; D_{α}v(x)|\frac{1}{ε}D_{3}v(x)) dx − ∫_{Ω} f · v dx − ∫_{Σ} g · v dH^{2}

on V_{ε} := \{v ∈ W^{1,p}(Ω; ℝ^{3}) : v(x) = (x_{α}, εx_{3}) on ∂ω × I\}. Denote by D_{i} = \frac{∂}{∂x_{i}} for i ∈ \{1, 2, 3\} and D_{α} = (D_{1}, D_{2}). In the sequel, we identify ℝ^{d×N} (resp. ℚ^{d×N}) with the space of real (resp. rational) d × N matrices. For all \overline{ξ} = (z_{1}|z_{2}) ∈ ℝ^{3×2} and z ∈ ℝ^{3}, (\overline{ξ}|z) is the matrix whose first two columns are z_{1} and z_{2} and whose last one is z. Denoting a quasi-minimizer of the rescaled energy by u_{ε}(x_{α}, x_{3}) := u(ε)(x_{α}, εx_{3}), we obtain

E_{ε}(u_{ε}) ⩽ inf_{v∈V_{ε}} E_{ε}(v) + h(ε).  (1.1)

Our aim is to study the asymptotic behavior of the equilibrium problem (1.1) as ε → 0 via a Γ-convergence method (we refer to Braides and Defranceschi [9], Braides [11] and Dal Maso [13] for a comprehensive treatment and bibliography on Γ-convergence).

The motivation for studying problem (1.1) comes from the work in [10] where the authors have established an abstract dimensional reduction variational convergence result in a general setting for a family of stored energies of the form W_{ε}(x; ξ) and derived specific characterizations for particular cases. In Section 3 of [10] a heterogeneous nonlinear membrane model is derived by Γ-convergence, and heterogeneity in the transverse direction is considered. Precisely, the authors treat the case where the stored energy density is of the form W (x_{3}; ξ), generalizing the previous work of Le Dret and Raoult in [17] who treated a homogeneous material, i.e. when W depends only in ξ. Later, Babadjian and Francfort [4] considered energies of the form W (x; ξ) with a general heterogeneity. Furthermore in Section 4 of [10], a 3D-2D analysis coupled with a homogenization in the in-plane direction is studied in the case where W_{ε}(x; ξ) = W (x_{3}, x_{α}/ε; ξ). Shu [22] also investigated similar problems, in the framework of martensitic materials, with different length scales for the film thickness and the material microstructure.

Here we propose to establish a dimensional reduction and homogenization result, where both scales are identical, by adding in the stored energy density an explicit dependence on the in-plane variable x_{α}. Namely, we assume that W_{ε}(x_{α}, x_{3}; ·) = W (x_{α}, x_{3}, x_{α}/ε; ·) for some function W : Ω × ℝ^{2} × ℝ^{3×3} → ℝ whose hypotheses will be introduced later.

Two features differentiate our approach from what is available in most of the literature in the subject. The first one is the use of a two-scale convergence argument (see Nguetseng [20, 21] and Allaire [2] for the notion and properties of two-scale convergence). The same argument was used by Baía and Fonseca in [5] in a pure homogenization context, i.e. without considering the dimensional reduction problem. The second feature is connected with the definition of the homogenized stored energy in which two independent variables are occurring: Firstly the integration variable y_{α} and secondly x_{α} on which this function is explicitly depending. To take into account this structure, we are led to decouple the oscillating variable from the in-plane variable. This is possible via an extension of Carathéodory functions argument in the lines of Babadjian and Francfort [4] in which the authors used Theorem 1 (Section 1.2) in Evans and Gariepy [15]. Due to the presence of the oscillating variable this last result can not be applied directly, and we are lead to extend it to suit our requirements.

For a comprehensive treatment on the homogenization of integral functionals via a Γ-limit approach, we refer to [9] and references therein. We will denote by L^{N} the N-dimensional Lebesgue measure in ℝ^{N} (in the sequel N will be equal to 2 or 3).

For each ε > 0 we define I_{ε} : L^{p}(Ω; ℝ^{3}) → \overline{ℝ} by

I_{ε}(u) := \{ \begin{matrix}∫_{Ω} W (x_{α}, x_{3}, \frac{x_{α}}{ε}; D_{α}u(x)|\frac{1}{ε}D_{3}u(x)) dx_{α} dx_{3} & if u ∈ W^{1,p}(Ω; ℝ^{3}), \\ +∞ & otherwise, \end{matrix}  (1.2)

with 1 < p < ∞, where we assume that W : Ω × ℝ^{2} × ℝ^{3×3} → ℝ satisfies the following hypotheses:

(H_{1}) W (x, ·; ·) is continuous for a.e. x ∈ Ω;

(H_{2}) W ( · , ·; ξ) is L^{3} ⊗ L^{2}-measurable for all ξ ∈ ℝ^{3×3}; (H_{3}) there exists 0 < β < +∞ such that

\frac{1}{β} |ξ|^{p} − β ⩽ W (x, y_{α}; ξ) ⩽ β(1 + |ξ|^{p}), for a.e. x ∈ Ω and for all (y_{α}, ξ) ∈ ℝ^{2} × ℝ^{3×3};

(H_{4}) W (x, ·; ξ) is Q′-periodic for a.e. x ∈ Ω and all ξ ∈ ℝ^{3×3}, where we denote by Q′ = (0, 1)^{2} the unit cube of ℝ^{2}.

Remark 1.1. We remark that due to hypothese (H_{1}) and (H_{2}) the function W is a Carathéodory integrand as W (x, ·; ·) is continuous a.e. x ∈ Ω and W (·, y_{α}; ξ) is measurable for all y_{α} ∈ ℝ^{2} and ξ ∈ ℝ^{3×3}. This implies (see e.g. Proposition 3.3 in [9] or Proposition 1.1, Chapter VIII in Ekeland and Temam [14]) that W is equivalent to a Borel function, that is there exist a Borel function W̃ such that W (x, ·; ·) = W̃ (x, ·; ·) for a.e. x ∈ Ω. As a consequence the integral in (1.2) is well defined. As noted by Allaire in [2], Section 5, the measurability of W in the pair (x, y_{α}) does not let us conclude that, for fixed ξ, the function x ↦ W (x, x_{α}/ε; ξ) is measurable. The continuity of W (x, y_{α}; ξ) in at least one of the variables x or y_{α} turns out to be sufficient to guarantee the measurability of this function. In the present paper, we decide to impose the continuity in the y_{α} variable. Note that we could also have considered W to be continuous in x and measurable in y_{α} but the proof of our main result does not hold anymore in this context.

As for notation, we will identify W^{1,p}(ω; ℝ^{3}) with the set of functions u ∈ W^{1,p}(Ω; ℝ^{3}) such that D_{3}u(x) = 0 for a.e. x ∈ Ω and we will use the notation Γ(L^{p}(Ω))-limit whenever we refer to the Γ-convergence with respect to the usual metric in L^{p}(Ω; ℝ^{3}). We prove the following main result.

Theorem 1.2. If W satisfies (H_{1}), (H_{2}), (H_{3}) and (H_{4}), then the family {I_{ε}}_{ε>0} Γ(L^{p}(Ω))-converges to the functional I_{hom} : L^{p}(Ω; ℝ^{3}) → \overline{ℝ} defined by

I_{hom}(u) := \{ \begin{matrix}2 ∫_{ω} W_{hom}(x_{α}; D_{α}u(x_{α})) dx_{α} & if u ∈ W^{1,p}(ω; ℝ^{3}), \\ +∞ & otherwise, \end{matrix}  (1.3)

where W_{hom} is given by

W_{hom}(x_{α}; \overline{ξ}) := lim_{T →+∞} inf_{φ} \{\frac{1}{2T^{2}} ∫_{(0,T)^{2}×I} W (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ(y)|D_{3}φ(y)) dy_{α}dy_{3} :
φ ∈ W^{1,p}((0, T)^{2} × I; ℝ^{3}), φ = 0 on ∂(0, T)^{2} × I}  (1.4)

for a.e. x_{α} ∈ ω and all \overline{ξ} ∈ ℝ^{3×2}.

We note that the p-coercivity condition in (H_{3}) ensures that the Γ-limit of I_{ε} remains unchanged if we consider the weak W^{1,p}-topology in place of the strong L^{p}-topology.

As a consequence of Theorem 1.2 we deduce the usual convergence of (quasi-)minimizers as stated in the following corollary.

Corollary 1.3. Let {u_{ε}} be the sequence of quasi-minimizers defined in (1.1). Then {u_{ε}} is weakly relatively compact in W^{1,p}(Ω; ℝ^{3}). Furthermore, any limit point u of this sequence is a solution of the minimization problem

min_{v−(x_{α},0)∈W_{0}^{1,p}(ω;ℝ^{3})} \{2 ∫_{ω} W_{hom}(x_{α}; D_{α}v(x_{α})) dx_{α} − ∫_{ω}(\overline{f} + g^{+} + g^{−})(x_{α}) · v(x_{α}) dx_{α}\} ,

where \overline{f} := \frac{1}{2} ∫_{I} f(·, x_{3}) dx_{3} and g^{±} := g(·, ±1).

This departs from the classical result (see Proposition 7.2 in [9]) due to the presence of the boundary conditions. This difficulty is overcome thanks to Remark 3.2 which says that we can prescribe the lateral boundary condition of the recovery sequence. We do not include the proof of this corollary because it is similar to that of Corollary 1.3 in Bouchitté, Fonseca and Mascarenhas [8].

The plan of this work is as follows: In Section 2 we will discuss some properties of W_{hom}, namely that it is well defined, proving that the limit on the right hand side of (1.4) exists, and that W_{hom}(x_{α}; ·) is continuous for a.e. x_{α} ∈ ω. Section 3 is devoted to the proof of our main result, Theorem 1.2. The starting point of our analysis is the Γ-limit integral representation result, Theorem 2.5 in [10]. Our objective is to identify the integrand, showing that is coincides (almost everywhere) with W_{hom}. An argument of two-scale convergence will be used to prove that the integrand of the Γ-limit is less than W_{hom} in Lemma 3.4, taking an oscillating sequence of test functions for the Γ-limit (see also Baía and Fonseca [5]). On the other hand, since the problem at fixed ε and the asymptotic problem are of different nature (a three-dimensional problem becoming a two-dimensional one), it appears a new difficulty to prove the converse inequality in Lemma 3.5. Indeed, arguing like in classical relaxation, we would use the Scorza-Dragoni Theorem (see assumptions (H_{1}) and (H_{2})) to find a compact set K ⊂ Ω, the complement of which has arbitrarily small Lebesgue measure and such that W is continuous on K × ℝ^{2} × ℝ^{3×3}. To localize our functional on small cubes so as to apply a uniform continuity argument, we fix x_{α}^{0} in ω and for all (x_{α}, x_{3}) ∈ K we must ensure that (x_{α}^{0}, x_{3}) ∈ K. But since K is compact, we cannot expect this to be true, and this argument fails. To overcome this difficulty (see also Babadjian and Francfort [4]), we need to replace W by a function which is (separately) continuous everywhere. That is the aim of Lemma 4.1 which provides a continuous extension of Carathéodory functions and will be proved in the Appendix in Section 4.

2 Preliminary results

In this section we will prove some properties of the stored energy W_{hom} that will be of use in the proof of Theorem 1.2.

Remark 2.1. To prove Theorem 1.2 we may assume, without loss of generality, that W is non negative. Indeed, in view of (H_{3}) it suffices to replace W by W + β.

We begin by showing that in the definition (1.4) of W_{hom} the limit as T → +∞ exists. The proof of this property is a direct consequence of a result due to Licht and Michaille [18], Theorem 3.1 (see also Lemma 4.3.6 in Bouchitté, Fonseca and Mascarenhas [7]). We introduce the following new condition :

(H′_{1}) W (x, y_{α}; ·) is continuous for a.e. x ∈ Ω and all y_{α} ∈ ℝ^{2}.

Remark 2.2. Note that (H_{1}) implies (H′_{1}). Furthermore, if W satisfies (H′_{1}) and (H_{2}), W is a Carathéodory function in the following sense : W (·, ·; ξ) is L^{3} ⊗ L^{2}-measurable for all ξ ∈ ℝ^{3×3} and W (x, y_{α}; ·) is continuous for L^{3} ⊗ L^{2}-a.e. (x, y_{α}) ∈ Ω × ℝ^{2}. As a consequence, there exists a Borel function W′ on Ω × ℝ^{2} × ℝ^{3×3} such that W (x, y_{α}; ·) = W′(x, y_{α}; ·) for L^{3} ⊗ L^{2}-a.e. (x, y_{α}) ∈ Ω × ℝ^{2}. Thus the integral in (1.4) is well defined. We insist on the fact that, in principle, W′ and W̃ (see Remark 1.1) need not to be equal.

Lemma 2.3. If W satisfies (H′_{1}), (H_{2}), (H_{3}) and (H_{4}), then

W_{hom}(x_{α}; \overline{ξ}) = lim_{T →+∞} inf_{φ} \{\frac{1}{2T^{2}} ∫_{(0,T)^{2}×I} W (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ(y)|D_{3}φ(y)) dy_{α}dy_{3} :
φ ∈ W^{1,p}((0, T)^{2} × I; ℝ^{3}), φ = 0 on ∂(0, T)^{2} × I}

exists for a.e. x_{α} ∈ ω and all \overline{ξ} ∈ ℝ^{3×2}.

Proof. Let x_{α} ∈ ω be such that (H′_{1}), (H_{3}) and (H_{4}) hold and let \overline{ξ} ∈ ℝ^{3×2}. Define μ : A(ℝ^{2}) → ℝ^{+} by

μ(A) := inf_{φ} \{\frac{1}{2} ∫_{A×I} W (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ(y)|D_{3}φ(y)) dy_{α}dy_{3} :
φ ∈ W^{1,p}(A × I; ℝ^{3}), φ = 0 on ∂A × I},

where A(ℝ^{2}) stands for the family of open subsets of ℝ^{2}.

In view of Remark 2.2, μ is well defined and, thanks to (H_{3}), it is a finite function. Moreover this set function satisfies the assumptions of Theorem 3.1 in Licht and Michaille [18]. Indeed firstly, by (H_{3}), μ(A) ⩽ β(1 + |\overline{ξ}|^{p}) L^{2}(A) for all A ∈ A(ℝ^{2}). Secondly, μ is subadditive, that is μ(C) ⩽ μ(A) + μ(B) for all A, B, C ∈ A(ℝ^{2}) with A ∩ B ≠ ∅ and \overline{C} = \overline{A} ∪ \overline{B}. Finally, by (H_{4}), for any i ∈ ℤ^{2}, μ(A + i) = μ(A) for all A ∈ A(ℝ^{2}). As a consequence the limit

lim_{T →+∞} \frac{μ((0, T)^{2})}{T^{2}} = W_{hom}(x_{α}; \overline{ξ})

exists. ■

Remark 2.4. It can be proved that the limit as T → +∞ in (1.4) can be replaced by an infimum taken for every T > 0 (see Braides and Defranceschi [9] or Baía and Fonseca [5]).

Now that W_{hom} is well defined, we will show that W_{hom}(x_{α}; ·) is continuous for a.e. x_{α} ∈ ω for later use in Theorem 1.2. To prove this property directly it seems that we would need a little bit more than only the continuity condition imposed on W (x, y_{α}; ·) (e.g. a p-Lipschitz condition). We remark that if W (x, y_{α}; ·) was quasiconvex, then by the p-growth condition (H_{3}), W (x, y_{α}; ·) would satisfy a p-Lipschitz condition (see Lemma 2.7 below). Since we do not want to a priori restrict too much the stored energy density, in order to compensate for this lack of regularity we prove first in Lemma 2.6 that the value of W_{hom} does not change if we replace W by its quasiconvexification QW (see Remark 2.5 below).

Remark 2.5. For a.e. x ∈ Ω, all y_{α} ∈ ℝ^{2} and all ξ ∈ ℝ^{3×3} define

QW (x, y_{α}; ξ) := [QW (x, y_{α}; ·)](ξ)

where QW (x, y_{α}; ·) stands for the usual quasiconvexification of W (x, y_{α}; ·). Then, the function QW (x, y_{α}; ·) is quasiconvex (see e.g. Dacorogna [12]) and by Remark 2.2 if W satisfies (H_{1}), (H_{2}), (H_{3}) and (H_{4}), so does QW except that QW (x, ·; ξ) may only be upper semicontinuous (as the infimum of continuous functions) for a.e. x ∈ Ω and all ξ ∈ ℝ^{3×3}. In particular, since QW satisfies (H′_{1}), (H_{2}), (H_{3}) and (H_{4}), by Lemma 2.3 it follows that

(QW)_{hom}(x_{α}; \overline{ξ}) = lim_{T →+∞} inf_{φ} \{\frac{1}{2T^{2}} ∫_{(0,T)^{2}×I} QW (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ(y)|D_{3}φ(y)) dy_{α}dy_{3} :
φ ∈ W^{1,p}((0, T)^{2} × I; ℝ^{3}), φ = 0 on ∂(0, T)^{2} × I}

exists for a.e. x_{α} ∈ ω and all \overline{ξ} ∈ ℝ^{3×2}.

Lemma 2.6. If W satisfies (H_{1}), (H_{2}), (H_{3}) and (H_{4}), then (QW)_{hom}(x_{α}; \overline{ξ}) = W_{hom}(x_{α}; \overline{ξ}) for a.e. x_{α} ∈ ω and all \overline{ξ} ∈ ℝ^{3×2}.

Proof. Let x_{α} ∈ ω be such that both (QW)_{hom}(x_{α}; ·) and W_{hom}(x_{α}; ·) are well defined. Since W ⩾ QW, we have W_{hom}(x_{α}; \overline{ξ}) ⩾ (QW)_{hom}(x_{α}; \overline{ξ}) for all \overline{ξ} ∈ ℝ^{3×2}. Let us prove now the converse inequality. Let \overline{ξ} ∈ ℝ^{3×2}. For each n > 0, let T_{n} ∈ ℕ and φ_{n} ∈ W^{1,∞}((0, T_{n})^{2} × I; ℝ^{3}) satisfying φ_{n} = 0 on ∂(0, T_{n})^{2} × I, be such that

(QW)_{hom}(x_{α}; \overline{ξ}) + \frac{1}{n} ⩾ \frac{1}{2T_{n}^{2}} ∫_{(0,T_{n})^{2}×I} QW (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ_{n}(y)|D_{3}φ_{n}(y)) dy_{α} dy_{3}.

The Lipschitz regularity of φ_{n} may be ensured because of the density of W^{1,∞}((0, T_{n})^{2}×I; ℝ^{3}) in W^{1,p}((0, T_{n})^{2}× I; ℝ^{3}) together with the p-growth condition (H_{3}). Thus

(QW)_{hom}(x_{α}; \overline{ξ}) ⩾ lim sup_{n→+∞} \frac{1}{2T_{n}^{2}} ∫_{(0,T_{n})^{2}×I} QW (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ_{n}(y)|D_{3}φ_{n}(y)) dy_{α} dy_{3}.  (2.1)

For each n ∈ ℕ fixed, by Acerbi-Fusco Relaxation Theorem (see Lemma III.1 and Statement III.7 in [1]) and Remark 2.1, there exists a sequence {φ_{n,k}}_{k} ⊂ W^{1,∞}((0, T_{n})^{2} × I; ℝ^{3}) satisfying φ_{n,k} = φ_{n} on ∂[(0, T_{n})^{2} × I] with φ_{n,k} −⇀_{k→+∞} φ_{n} and such that

\begin{matrix}\frac{1}{2T_{n}^{2}} ∫_{(0,T_{n})^{2}×I} QW (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ_{n}(y)|D_{3}φ_{n}(y)) dy_{α} dy_{3} \\ = lim_{k→+∞} \frac{1}{2T_{n}^{2}} ∫_{(0,T_{n})^{2}×I} W (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ_{n,k}(y)|D_{3}φ_{n,k}(y)) dy_{α} dy_{3}. \end{matrix}

From (2.1) we have

(QW)_{hom}(x_{α}; \overline{ξ}) ⩾ lim sup_{n→+∞} lim sup_{k→+∞} \frac{1}{2T_{n}^{2}} ∫_{(0,T_{n})^{2}×I} W (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ_{n,k}(y)|D_{3}φ_{n,k}(y)) dy_{α} dy_{3}
⩾ lim sup_{n→∞} inf_{φ} \{\frac{1}{2T_{n}^{2}} ∫_{(0,T_{n})^{2}×I} W (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ(y)|D_{3}φ(y)) dy_{α}dy_{3} :
φ ∈ W^{1,p}((0, T_{n})^{2} × I; ℝ^{3}), φ = 0 on ∂(0, T_{n})^{2} × I}
= W_{hom}(x_{α}; \overline{ξ}).
■

We are now in position to prove the continuity of W_{hom} in its second variable :

Lemma 2.7. Let W satisfying (H_{1}), (H_{2}), (H_{3}) and (H_{4}), then W_{hom}(x_{α}; ·) is continuous on ℝ^{3×2} for a.e. x_{α} ∈ ω.

Proof. We observe that by the p-growth condition in (H_{3}) and Remark 2.5, QW satisfies a p-Lipschitz condition (see Marcellini [19]): There exists β > 0 such that for all y_{α} ∈ ℝ^{2} and a.e. x ∈ Ω,

|QW (x, y_{α}; ξ_{1}) − QW (x, y_{α}; ξ_{2})| ⩽ β(1 + |ξ_{1}|^{p−1} + |ξ_{2}|^{p−1})|ξ_{1} − ξ_{2}|, ξ_{1}, ξ_{2} ∈ ℝ^{3×3}.  (2.2)

Take x_{α} ∈ ω such that both (QW)_{hom}(x_{α}; ·) and W_{hom}(x_{α}; ·) are well defined. By Lemma 2.6 we have (QW)_{hom}(x_{α}; ·) = W_{hom}(x_{α}; ·). Given \overline{ξ} ∈ ℝ^{3×2} let \overline{ξ}_{n} → \overline{ξ} in ℝ^{3×2}. From the definition of W_{hom}(x_{α}; \overline{ξ}), for fixed δ > 0 choose T ∈ ℕ and φ ∈ W^{1,p}((0, T)^{2} × I; ℝ^{3}), φ = 0 on ∂(0, T)^{2} × I, such that

W_{hom}(x_{α}; \overline{ξ}) + δ ⩾ \frac{1}{2T^{2}} ∫_{(0,T)^{2}×I} W (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ(y)|D_{3}φ(y)) dy_{α}dy_{3}.  (2.3)

Therefore, Remark 2.4 yields

lim sup_{n→+∞} W_{hom}(x_{α}; \overline{ξ}_{n}) ⩽ lim sup_{n→+∞} \frac{1}{2T^{2}} ∫_{(0,T)^{2}×I} W (x_{α}, y_{3}, y_{α}; \overline{ξ}_{n} + D_{α}φ(y)|D_{3}φ(y)) dy_{α}dy_{3}
= \frac{1}{2T^{2}} ∫_{(0,T)^{2}×I} W (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ(y)|D_{3}φ(y)) dy_{α}dy_{3}

due to hypothesis (H_{1}), the p-growth condition in (H_{3}) and Lebesgue’s Dominated Convergence Theorem. So by (2.3) and letting δ → 0 we conclude that

lim sup_{n→+∞} W_{hom}(x_{α}; \overline{ξ_{n}}) ⩽ W_{hom}(x_{α}; \overline{ξ}).  (2.4)

Similarly, for each n ∈ ℕ consider T_{n} ∈ ℕ (T_{n} ↗ +∞) and φ_{n} ∈ W^{1,p}((0, T_{n})^{2} ×I; ℝ^{3}), φ_{n} = 0 on ∂(0, T_{n})^{2} ×I, such that

W_{hom}(x_{α}; \overline{ξ}_{n}) + \frac{1}{n} ⩾ \frac{1}{2T_{n}^{2}} ∫_{(0,T_{n})^{2}×I} QW (x_{α}, y_{3}, y_{α}; \overline{ξ}_{n} + D_{α}φ_{n}(y)|D_{3}φ_{n}(y)) dy_{α}dy_{3}
= \frac{1}{2} ∫_{Q′×I} QW (x_{α}, y_{3}, T_{n}y_{α}; \overline{ξ}_{n} + D_{α}φ_{n}(T_{n}y_{α}, y_{3})|D_{3}φ_{n}(T_{n}y_{α}, y_{3})) dy_{α}dy_{3}
= \frac{1}{2} ∫_{Q′×I} QW (x_{α}, y_{3}, T_{n}y_{α}; \overline{ξ}_{n} + D_{α}ψ_{n}(y)|T_{n}D_{3}ψ_{n}(y)) dy_{α}dy_{3},

after a change of variables and where ψ_{n}(y) := \frac{1}{T_{n}} φ_{n}(T_{n}y_{α}, y_{3}). Clearly the function ψ_{n} belongs to W^{1,p}(Q′ × I; ℝ^{3}) and ψ_{n} = 0 on ∂Q′×I. By the p-coercivity hypothesis in (H_{3}) and (2.4), the sequence {(D_{α}ψ_{n}|T_{n}D_{3}ψ_{n})} is bounded in L^{p}(Q′ × I; ℝ^{3×3}) uniformly in n. We can write that

lim inf_{n→+∞} ∫_{Q′×I} QW (x_{α}, y_{3}, T_{n}y_{α}; \overline{ξ}_{n} + D_{α}ψ_{n}(y)|T_{n}D_{3}ψ_{n}(y)) dy_{α}dy_{3}

⩾ lim inf_{n→+∞} ∫_{Q′×I} [ QW (x_{α}, y_{3}, T_{n}y_{α}; \overline{ξ}_{n} + D_{α}ψ_{n}(y)|T_{n}D_{3}ψ_{n}(y))

−QW (x_{α}, y_{3}, T_{n}y_{α}; \overline{ξ} + D_{α}ψ_{n}(y)|T_{n}D_{3}ψ_{n}(y)) ] dy_{α} dy_{3}

+ lim inf_{n→+∞} ∫_{Q′×I} QW (x_{α}, y_{3}, T_{n}y_{α}; \overline{ξ} + D_{α}ψ_{n}(y)|T_{n}D_{3}ψ_{n}(y)) dy_{α}dy_{3}.

Using (2.2), Hölder inequality, the fact that \{∥(D_{α}ψ_{n}|T_{n}D_{3}ψ_{n})∥_{L^{p}(Q′×I;ℝ^{3×3})}\} is bounded and \overline{ξ}_{n} → \overline{ξ}, we obtain

lim inf_{n→+∞} ∫_{Q′×I} [ QW (x_{α}, y_{3}, T_{n}y_{α}; \overline{ξ}_{n} + D_{α}ψ_{n}(y)|T_{n}D_{3}ψ_{n}(y))
−QW (x_{α}, y_{3}, T_{n}y_{α}; \overline{ξ} + D_{α}ψ_{n}(y)|T_{n}D_{3}ψ_{n}(y)) ] dy_{α} dy_{3} = 0,

and consequently

lim inf_{n→+∞} W_{hom}(x_{α}; \overline{ξ}_{n}) ⩾ lim inf_{n→+∞} \frac{1}{2} ∫_{Q′×I} QW (x_{α}, y_{3}, T_{n}y_{α}; \overline{ξ} + D_{α}ψ_{n}(y)|T_{n}D_{3}ψ_{n}(y)) dy_{α}dy_{3}
= lim inf_{n→+∞} \frac{1}{2T_{n}^{2}} ∫_{(0,T_{n})^{2}×I} QW (x_{α}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ_{n}(y)|D_{3}φ_{n}(y)) dy_{α}dy_{3}
⩾ (QW)_{hom}(x_{α}; \overline{ξ})
= W_{hom}(x_{α}; \overline{ξ}).  (2.5)

From (2.4) and (2.5), we conclude that W_{hom}(x_{α}; ·) is continuous at \overline{ξ}. ■

3 Proof of Theorem 1.2

We start by localizing our functionals. Representing by A(ω) the class of all open subsets of ω, define I_{ε} : L^{p}(Ω; ℝ^{3}) × A(ω) → \overline{ℝ} by

I_{ε}(u; A) := \{ \begin{matrix}∫_{A×I} W (x_{α}, x_{3}, \frac{x_{α}}{ε}; D_{α}u(x)|\frac{1}{ε}D_{3}u(x)) dx_{α} dx_{3} & if u ∈ W^{1,p}(A × I; ℝ^{3}), \\ +∞ & otherwise. \end{matrix}

We will prove that the family of functionals {I_{ε}(·; A)}_{ε>0} Γ-converges with respect to the L^{p}(A × I; ℝ^{3})- topology to the functional I_{hom}(·; A) : L^{p}(Ω; ℝ^{3}) → \overline{ℝ}

I_{hom}(u; A) := \{ \begin{matrix}2 ∫_{A} W_{hom}(x_{α}; D_{α}u(x_{α})) dx_{α} & if u ∈ W^{1,p}(A; ℝ^{3}), \\ +∞ & otherwise, \end{matrix}  (3.1)

for all A ∈ A(ω). As a consequence, taking A = ω yields Theorem 1.2.

For any A ∈ A(ω) and any sequence \{ε_{j}\} ↘ 0^{+}, consider I_{\{ε_{j}\}}(·; A) : L^{p}(Ω; ℝ^{3}) → \overline{ℝ} the Γ-lower limit of {I_{ε_{j}}(·; A)}_{j∈ℕ},

I_{{ε_{j}}}(u; A) := inf_{{u_{j}}} {lim inf_{j→+∞} I_{ε_{j}}(u_{j}; A) : u_{j} → u in L^{p}(A × I; ℝ^{3})} .  (3.2)

Remark 3.1. In view of the coercivity condition (H_{4}), for all A ∈ A(ω) we have that I_{{ε_{j}}}(u; A) = +∞ whenever u ∈ L^{p}(Ω; ℝ^{3}) \ W^{1,p}(A; ℝ^{3}), hence our objective is to characterize I_{{ε_{j}}}(u; A) for u ∈ W^{1,p}(A; ℝ^{3}).

By virtue of Remark 3.1, together with Theorem 2.5 in Braides, Fonseca and Francfort [10], it follows that every sequence {ε_{j}} admits a subsequence {ε_{j_{n}}} ≡ {ε_{n}} such that I_{{ε_{n}}}( ·; A) defined in (3.2) is the Γ(L^{p}(A×I))- limit of {I_{ε_{n}}( ·; A)}_{n∈ℕ} for all A ∈ A(ω). Further there exists a Carathéodory function W_{{ε_{n}}} : ω × ℝ^{3×2} → ℝ such that

I_{{ε_{n}}}(u; A) = 2 ∫_{A} W_{{ε_{n}}}(x_{α}; D_{α}u(x_{α})) dx_{α},  (3.3)

for all A ∈ A(ω) and all u ∈ W^{1,p}(A; ℝ^{3}).

Our aim is to show that I_{{ε_{n}}}(·; A) = I_{hom}(·; A) on W^{1,p}(A; ℝ^{3}) for all A ∈ A(ω). Given A ∈ A(ω), in view of the integral representation (3.3) and (3.1), it is enough to show that W_{\{ε_{n}\}}(x_{α}; \overline{ξ}) = W_{hom}(x_{α}; \overline{ξ}) for a.e. x_{α} ∈ A and all \overline{ξ} ∈ ℝ^{3×2}, and thus to work with affine functions instead of general Sobolev functions. We will prove that W_{\{ε_{n}\}}(x_{α}; \overline{ξ}) = W_{hom}(x_{α}; \overline{ξ}) for a.e. x_{α} ∈ ω and all \overline{ξ} ∈ ℝ^{3×2}.

Remark 3.2. Lemma 2.6 of Braides, Fonseca and Francfort [10] implies that I_{{ε_{j}}}(u; A) is unchanged if the approximating sequences {u_{j}} are constrained to match the lateral boundary condition of their target, i.e. u_{j} ≡ u on ∂A × I.

From now onward, {ε_{n}} will denote a subsequence of {ε_{j}} for which the Γ(L^{p}(A×I))-limit of {I_{ε_{n}}(·; A)}_{n∈ℕ} exists and coincides with I_{{ε_{n}}}(·; A) for all A ∈ A(ω).

For each T > 0 consider S_{T} a countable set of functions in C^{∞}([0, T]^{2} × [−1, 1]; ℝ^{3}) that is dense in

W_{T} = {φ ∈ W^{1,p}((0, T)^{2} × I; ℝ^{3}) : φ = 0 on ∂(0, T)^{2} × I}.

Definition 3.3. Let L be the set of Lebesgue points x_{α}^{0} for all functions

W_{\{ε_{n}\}}(·; \overline{ξ}), W_{hom}(·; \overline{ξ})  (3.4)

and

x_{α} ↦ ∫_{Q′×I} W (x_{α}, y_{3}, T y_{α}; \overline{ξ} + D_{α}φ(T y_{α}, y_{3})|D_{3}φ(T y_{α}, y_{3})) dy_{α} dy_{3},  (3.5)

with T ∈ ℕ, φ ∈ S_{T} and \overline{ξ} ∈ ℚ^{3×2}, and for which W_{hom}(x_{α}^{0}; ·) is well defined.

We have that L^{2}(ω \ L) = 0. Given x_{α}^{0} ∈ L, we denote by Q′(x_{α}^{0}, δ) the cube in ℝ^{2} centered in x_{α}^{0} and of side length δ > 0 where δ is small enough so that Q′(x_{α}^{0}, δ) ∈ A(ω).

To prove that W_{\{ε_{n}\}}(x_{α}; \overline{ξ}) = W_{hom}(x_{α}; \overline{ξ}) for a.e. x_{α} ∈ ω and all \overline{ξ} ∈ ℝ^{3×2} we first show in Lemmas 3.4 and 3.5 below that both functions coincide on L × ℚ^{3×2}. The general case will only be treated at the end of that section using the Carathéodory property of both integrands.

Fix \overline{ξ} ∈ ℚ^{3×2} and set v(x) := \overline{ξ} · x_{α}. By (3.3) and (3.4)

W_{\{ε_{n}\}}(x_{α}^{0}; \overline{ξ}) = lim_{δ→0} \frac{1}{δ^{2}} ∫_{Q′(x_{α}^{0},δ)} W_{\{ε_{n}\}}(x_{α}; \overline{ξ}) dx_{α}

= lim_{δ→0} \frac{I_{\{ε_{n}\}}(v; Q′(x_{α}^{0}, δ))}{2δ^{2}}.  (3.6)

Lemma 3.4. W_{\{ε_{n}\}}(x_{α}^{0}; \overline{ξ}) ⩽ W_{hom}(x_{α}^{0}; \overline{ξ}) for all x_{α}^{0} ∈ L and all \overline{ξ} ∈ ℚ^{3×2}.

Proof. Given k ∈ ℕ, let T_{k} ∈ ℕ and φ_{k} ∈ S_{T_{k}} with φ_{k} = 0 on ∂(0, T_{k})^{2} × I, be such that

W_{hom}(x_{α}^{0}; \overline{ξ}) + \frac{1}{k} ⩾ \frac{1}{2T_{k}^{2}} ∫_{(0,T_{k})^{2}×I} W (x_{α}^{0}, y_{3}, y_{α}; \overline{ξ} + D_{α}φ_{k}(y)|D_{3}φ_{k}(y)) dy.

This is possible because of the continuity properties (H_{1}) of W, the growth conditions (H_{3}) and the density of S_{T_{k}} in W_{T_{k}}. Extend φ_{k} periodically with period T_{k} to ℝ^{2}×I. For x ∈ ℝ^{2}×I, define u_{n}^{k}(x) := \overline{ξ}·x_{α}+ε_{n}φ_{k}(\frac{x_{α}}{ε_{n}} , x_{3}). For fixed k, u_{n}^{k} → v in L^{p}(Q′(x_{α}^{0}, δ) × I; ℝ^{3}) as n → ∞, hence, by (3.6)

W_{\{ε_{n}\}}(x_{α}^{0}; \overline{ξ}) ⩽ lim inf_{δ→0} lim inf_{n→+∞} \frac{1}{2δ^{2}} ∫_{Q′(x_{α}^{0},δ)×I} W (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; D_{α}u_{n}^{k}|\frac{1}{ε_{n}} D_{3}u_{n}^{k}) dx_{α}dx_{3}

= lim inf_{δ→0} lim inf_{n→+∞} \frac{1}{2δ^{2}} ∫_{Q′(x_{α}^{0},δ)×I} W (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}φ_{k} (\frac{x_{α}}{ε_{n}} , x_{3}) |D_{3}φ_{k} (\frac{x_{α}}{ε_{n}} , x_{3})) dx_{α}dx_{3}.

Define

h_{k}(x_{α}, y_{α}) := ∫_{−1}^{1} W (x_{α}, x_{3}, T_{k}y_{α}; \overline{ξ} + D_{α}φ_{k}(T_{k}y_{α}, x_{3})|D_{3}φ_{k}(T_{k}y_{α}, x_{3}))dx_{3}, for x_{α} ∈ ω and y_{α} ∈ ℝ^{2}.

The continuity of W with respect to y_{α}, its measurability and periodicity properties, and the fact that T_{k} ∈ ℕ lead us to conclude that the function h_{k} ∈ L^{1}(Q′(x_{α}^{0}, δ); C_{per}(Q′)) for fixed δ > 0, where C_{per}(Q′) denotes the space of Q′-periodic and continuous functions defined on ℝ^{2} (see Lemma 5.3 in Allaire [2]). Lemma 5.2 in [2] together with Fubini’s Theorem yields to

lim_{n→+∞} ∫_{Q′(x_{α}^{0},δ)×I} W (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}φ_{k} (\frac{x_{α}}{ε_{n}} , x_{3}) |D_{3}φ_{k} (\frac{x_{α}}{ε_{n}} , x_{3})) dx_{α}dx_{3}

= lim_{n→+∞} ∫_{Q′(x_{α}^{0},δ)} h_{k} (x_{α}, \frac{x_{α}}{T_{k}ε_{n}} ) dx_{α}
= ∫_{Q′(x_{α}^{0},δ)} ∫_{Q′} h_{k}(x_{α}, y_{α}) dy_{α} dx_{α}
= ∫_{Q′(x_{α}^{0},δ)} ∫_{Q′×I} W (x_{α}, x_{3}, T_{k}y_{α}; \overline{ξ} + D_{α}φ_{k}(T_{k}y_{α}, x_{3})|D_{3}φ_{k}(T_{k}y_{α}, x_{3}))dy_{α} dx_{3} dx_{α}.

Using (3.5) we have

W_{\{ε_{n}\}}(x_{α}^{0}; \overline{ξ})

⩽ lim inf_{δ→0} \frac{1}{2δ^{2}} ∫_{Q′(x_{α}^{0},δ)} ∫_{Q′×I} W (x_{α}, x_{3}, T_{k}y_{α}; \overline{ξ} + D_{α}φ_{k}(T_{k}y_{α}, x_{3})|D_{3}φ_{k}(T_{k}y_{α}, x_{3}))dy_{α} dx_{3}dx_{α}

= \frac{1}{2} ∫_{Q′×I} W (x_{α}^{0}, x_{3}, T_{k}y_{α}; \overline{ξ} + D_{α}φ_{k}(T_{k}y_{α}, x_{3})|D_{3}φ_{k}(T_{k}y_{α}, x_{3}))dy_{α} dx_{3}

⩽ W_{hom}(x_{α}^{0}; \overline{ξ}) + \frac{1}{k} .

Letting k → +∞ we assert the claim.

■

Note that the same proof could be used to prove Lemma 2.5 in Babadjian and Francfort [4].

Lemma 3.5. W_{\{ε_{n}\}}(x_{α}^{0}; \overline{ξ}) ⩾ W_{hom}(x_{α}^{0}; \overline{ξ}) for all x_{α}^{0} ∈ L and all \overline{ξ} ∈ ℚ^{3×2}.

Proof. Let {v_{n}} ⊂ W^{1,p}(Q′(x_{α}^{0}, δ) × I; ℝ^{3}) be a recovering sequence for the Γ-limit, i.e.

v_{n} → 0 in L^{p}(Q′(x_{α}^{0}, δ) × I; ℝ^{3})

and

I_{\{ε_{n}\}}(v; Q′(x_{α}^{0}, δ)) = lim_{n→+∞} ∫_{Q′(x_{α}^{0},δ)×I} W (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}v_{n}|\frac{1}{ε_{n}} D_{3}v_{n}) dx_{α}dx_{3}.

According to Theorem 1.1 in Bocea and Fonseca [6], there exists a subsequence of {ε_{n}} (not relabelled) and a sequence {u_{n}} ⊂ W^{1,p}(Q′(x_{α}^{0}, δ) × I; ℝ^{3}) such that, upon setting E_{n} := {x ∈ Q′(x_{α}^{0}, δ) × I : u_{n}(x) = v_{n}(x)}, then

\{ \begin{matrix}u_{n} → 0 in L^{p}(Q′(x_{α}^{0}, δ) × I; ℝ^{3}), \\ \{|(D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n})|^{p}\} is equi-integrable, \\ lim_{n→+∞} L^{3}([Q′(x_{α}^{0}, δ) × I] \ E_{n}) = 0. \end{matrix}  (3.7)

Thus, in view of the p-growth condition (H_{3}) together with (3.7) and Remark 2.1,

I_{\{ε_{n}\}}(v; Q′(x_{α}^{0}, δ)) ⩾ lim sup_{n→+∞} ∫_{E_{n}} W (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n}) dx_{α}dx_{3}
= lim sup_{n→+∞} ∫_{Q′(x_{α}^{0},δ)×I} W (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n}) dx_{α}dx_{3}

− lim sup_{n→+∞} ∫_{[Q′(x_{α}^{0},δ)×I]\E_{n}} W (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n}) dx_{α}dx_{3}

⩾ lim sup_{n→+∞} ∫_{Q′(x_{α}^{0},δ)×I} W (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n}) dx_{α}dx_{3}.

For any h ∈ ℕ, we split Q′(x_{α}^{0}, δ) into h^{2} disjoints cubes Q′_{i,h} of side length δ/h so that Q′(x_{α}^{0}, δ) = ⋃_{i=1}^{h^{2}} Q′_{i,h} and

I_{\{ε_{n}\}}(v; Q′(x_{α}^{0}, δ)) ⩾ lim sup_{h→+∞} lim sup_{n→+∞} ∑_{i=1}^{h^{2}} ∫_{Q′_{i,h}×I} W (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n}) dx.  (3.8)

For every η > 0 and λ > 0, let K_{η} ⊂ Ω and W^{η,λ} be given by Lemma 4.1 below (with N = d = 3, m = 2 and f = W). Then

L^{3}(Ω \ K_{η}) < η.  (3.9)

On the other hand, define

R_{n}^{λ} := \{x ∈ Q′(x_{α}^{0}, δ) × I : |(\overline{ξ} + D_{α}u_{n}(x)|\frac{1}{ε_{n}} D_{3}u_{n}(x))| ⩽ λ\} .

Chebyshev’s inequality implies that there exists a constant C > 0 – which does not depend on n or λ – such that

L^{3}([Q′(x_{α}^{0}, δ) × I] \ R_{n}^{λ}) < \frac{C}{λ^{p}} .  (3.10)

Since W and W^{η,λ} coincide on K_{η} × ℝ^{2} × \overline{B}(0, λ), where in the sequel the set \overline{B}(0, λ) stands for the closed ball {ξ ∈ ℝ^{3×3} : |ξ| ⩽ λ} of ℝ^{3×3}, we get in view of (3.8)

I_{{ε_{n}}}(v; Q′(x_{α}^{0}, δ)) ⩾

lim sup_{λ→+∞} lim sup_{η→0} lim sup_{h→+∞} lim sup_{n→+∞} ∑_{i=1}^{h^{2}} ∫_{[Q′_{i,h}×I]∩R_{n}^{λ}∩K_{η}} W^{η,λ} (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n}) dx.

By virtue of (4.1) below and (3.9),

∑_{i=1}^{h^{2}} ∫_{([Q′_{i,h}×I]∩R_{n}^{λ})\K_{η}} W^{η,λ} (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n}) dx ⩽ β(1 + λ^{p})η ⟶_{η→0} 0,

uniformly in (n, h), so that

I_{{ε_{n}}}(v; Q′(x_{α}^{0}, δ)) ⩾

lim sup_{λ→+∞} lim sup_{η→0} lim sup_{h→+∞} lim sup_{n→+∞} ∑_{i=1}^{h^{2}} ∫_{[Q′_{i,h}×I]∩R_{n}^{λ}} W^{η,λ} (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n}) dx.

Fix y_{α} ∈ Q′. Since W^{η,λ}(·, y_{α}; ·) is continuous, it is uniformly continuous on \overline{Ω} × \overline{B}(0, λ), and we define the modulus of continuity ω_{η,λ} : Q′ × ℝ^{+} → ℝ^{+} by

ω_{η,λ}(y_{α}, t) := sup_{(x,ξ), (x′,ξ′)∈\overline{Ω}×\overline{B}(0,λ)}\{|W^{η,λ}(x, y_{α}; ξ) − W^{η,λ}(x′, y_{α}; ξ′)| : |(x; ξ) − (x′; ξ′)| ⩽ t\}.

Then

ω_{η,λ}(·, t) is lower semicontinuous for all t ∈ ℝ^{+},
ω_{η,λ}(y_{α}, ·) is continuous and increasing for all y_{α} ∈ Q′,
ω_{η,λ}(y_{α}, 0) = 0 for all y_{α} ∈ Q′,

and

|W^{η,λ}(x, y_{α}; ξ) − W^{η,λ}(x′, y_{α}; ξ′)| ⩽ ω_{η,λ}(y_{α}, |x − x′| + |ξ − ξ′|) for all (x, ξ), (x′, ξ′) ∈ \overline{Ω} × \overline{B}(0, λ). (3.11)

The first property is a consequence of the fact that the supremum of continuous functions is lower semicontinuous, while the other ones are classical properties of moduli of continuity.

For all t ∈ ℝ^{+}, we extend ω_{η,λ}(·, t) to ℝ^{2} by Q′-periodicity. Since W^{η,λ}(x, ·; ξ) is Q′-periodic, inequality (3.11) holds for all y_{α} ∈ ℝ^{2}. Consequently, for every (x_{α}, x_{3}) ∈ [Q′_{i,h} × I] ∩ R_{n}^{λ} and every x′_{α} ∈ Q′_{i,h},

|W^{η,λ} (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3}))

−W^{η,λ} (x′_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3}))|

⩽ ω_{η,λ} (\frac{x_{α}}{ε_{n}} , |x_{α} − x′_{α}|) ⩽ ω_{η,λ} (\frac{x_{α}}{ε_{n}} , \frac{\sqrt{2}δ}{h} ) .

We get, after integration in (x_{α}, x_{3}, x′_{α}) and summation,

∑_{i=1}^{h^{2}} \frac{h^{2}}{δ^{2}} ∫_{Q′_{i,h}} \{∫_{R_{n}^{λ}∩[Q′_{i,h}×I]} |W^{η,λ} (x_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3}))

−W^{η,λ} (x′_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3}))| dx\} dx′_{α}

⩽ 2 ∫_{Q′(x_{α}^{0},δ)} ω_{η,λ} (\frac{x_{α}}{ε_{n}} , \frac{\sqrt{2}δ}{h} ) dx_{α}.

Riemann-Lebesgue’s Lemma applied to the Q′-periodic function ω_{η,λ}( · , \sqrt{2}δ/h) yields,

lim_{n→+∞} 2 ∫_{Q′(x_{α}^{0},δ)} ω_{η,λ} (\frac{x_{α}}{ε_{n}} , \frac{\sqrt{2}δ}{h} ) dx_{α} = 2δ^{2} ∫_{Q′} ω_{η,λ} (x_{α}, \frac{\sqrt{2}δ}{h} ) dx_{α},

and by Beppo-Levi’s Monotone Convergence Theorem

lim_{h→+∞} 2δ^{2} ∫_{Q′} ω_{η,λ} (x_{α}, \frac{\sqrt{2}δ}{h} ) dx_{α} = 0.

Hence, denoting by lim sup_{λ,η,h,n} the successive lim sup_{λ→+∞} lim sup_{η→0} lim sup_{h→+∞} lim sup_{n→+∞},

I_{{ε_{n}}}(v; Q′(x_{α}^{0}, δ)) ⩾

lim sup_{λ,η,h,n} ∑_{i=1}^{h^{2}} \frac{h^{2}}{δ^{2}}∫_{Q′_{i,h}}\{∫_{[Q′_{i,h}×I]∩R_{n}^{λ}}W^{η,λ}(x′_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3})) dx_{α}dx_{3}\} dx′_{α}.

Define the following sets which depend on all parameters (η, λ, i, h, n) :

\{ \begin{matrix}T := \{(x′_{α}, x_{α}, x_{3}) ∈ Q′_{i,h} × Q′_{i,h} × I : (x′_{α}, x_{3}) ∈ K_{η} and (x_{α}, x_{3}) ∈ R_{n}^{λ}\}, \\ T_{1} := \{(x′_{α}, x_{α}, x_{3}) ∈ Q′_{i,h} × Q′_{i,h} × I : (x′_{α}, x_{3}) ∉ K_{η} and (x_{α}, x_{3}) ∈ R_{n}^{λ}\}, \\ T_{2} := \{(x′_{α}, x_{α}, x_{3}) ∈ Q′_{i,h} × Q′_{i,h} × I : (x_{α}, x_{3}) ∉ R_{n}^{λ}\}, \end{matrix}

and note that Q′_{i,h} × Q′_{i,h} × I = T ∪ T_{1} ∪ T_{2}. Since W (·, y_{α}; ·) and W^{η,λ}(·, y_{α}; ·) coincide on K_{η} × \overline{B}(0, λ), we have

I_{{ε_{n}}}(v; Q′(x_{α}^{0}, δ))

⩾ lim sup_{λ,η,h,n} ∑_{i=1}^{h^{2}} \frac{h^{2}}{δ^{2}} ∫_{T} W^{η,λ} (x′_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3})) dx dx′_{α}

= lim sup_{λ,η,h,n} ∑_{i=1}^{h^{2}} \frac{h^{2}}{δ^{2}} ∫_{T} W (x′_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3})) dx dx′_{α}. (3.12)

We will prove that the corresponding terms over T_{1} and T_{2} are zero. Indeed, in view of (3.9) and the p-growth condition (H_{3}),

∑_{i=1}^{h^{2}} \frac{h^{2}}{δ^{2}} ∫_{T_{1}} W (x′_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3})) dx dx′_{α}

⩽ ∑_{i=1}^{h^{2}} \frac{h^{2}}{δ^{2}} L^{2}(Q′_{i,h}) L^{3}([Q′_{i,h} × I] \ K_{η}) β(1 + λ^{p})

< β(1 + λ^{p})η ⟶_{η→0} 0,  (3.13)

uniformly in (n, h). The bound from above in (H_{3}), the equi-integrability of \{|(D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n})|^{p}\} and (3.10) imply that

∑_{i=1}^{h^{2}} \frac{h^{2}}{δ^{2}} ∫_{T_{2}} W (x′_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3})) dx dx′_{α}

⩽ ∑_{i=1}^{h^{2}} \frac{h^{2}}{δ^{2}} L^{2}(Q′_{i,h}) β ∫_{[Q′_{i,h}×I]\R_{n}^{λ}} (1 + |(D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n})|^{p}) dx

= β ∫_{[Q′(x_{α}^{0},δ)×I]\R_{n}^{λ}} (1 + |(D_{α}u_{n}|\frac{1}{ε_{n}} D_{3}u_{n})|^{p}) dx ⟶_{λ→+∞} 0,  (3.14)

uniformly in (η, n, h). Thus, in view of (3.12), (3.13), (3.14), Fatou’s Lemma yields

I_{{ε_{n}}}(v; Q′(x_{α}^{0}, δ))

⩾lim sup_{h→+∞} lim sup_{n→+∞} ∑_{i=1}^{h^{2}} \frac{h^{2}}{δ^{2}} ∫_{Q′_{i,h}}∫_{Q′_{i,h}×I}W (x′_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3}))dx_{α}dx_{3}dx′_{α}

⩾lim sup_{h→+∞} ∑_{i=1}^{h^{2}} \frac{h^{2}}{δ^{2}}∫_{Q′_{i,h}} lim inf_{n→+∞} ∫_{Q′_{i,h}×I}W (x′_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3}))dx_{α}dx_{3}dx′_{α}.

Fix x′_{α} ∈ Q′_{i,h} such that W_{hom}(x′_{α}; \overline{ξ}) is well defined and set Z(x; ξ) := W (x′_{α}, x_{3}, x_{α}; ξ). It is easy to check that Z is a Carathéodory integrand hence, applying Theorem 4.2 of Braides, Fonseca and Francfort [10], we get since u_{n} → 0 in L^{p}(Q′(x_{α}^{0}, δ) × I; ℝ^{3}),

2\frac{δ^{2}}{h^{2}} \overline{Z}(\overline{ξ}) ⩽ lim inf_{n→+∞} ∫_{Q′(x_{α}^{0},δ)×I} Z (\frac{x_{α}}{ε_{n}} , x_{3}; \overline{ξ} + D_{α}u_{n}(x)|\frac{1}{ε}D_{3}u_{n}(x)) dx,

where

\overline{Z}(\overline{ξ}) := inf_{T >0, φ} \{ ∫_{(0,T)^{2}×I} Z(x; \overline{ξ} + D_{α}φ(x)|D_{3}φ(x)) dx :
φ ∈ W^{1,p}((0, T)^{2} × I; ℝ^{3}), φ = 0 on ∂(0, T)^{2} × I}.

In view of the previous formula together with (1.4) and Remark 2.4, we have that \overline{Z}(\overline{ξ}) = W_{hom}(x′_{α}; \overline{ξ}). Then

lim inf_{n→+∞} ∫_{Q′_{i,h}×I} W (x′_{α}, x_{3}, \frac{x_{α}}{ε_{n}}; \overline{ξ} + D_{α}u_{n}(x_{α}, x_{3})|\frac{1}{ε_{n}} D_{3}u_{n}(x_{α}, x_{3})) dx_{α}dx_{3} ⩾ \frac{2δ^{2}}{h^{2}} W_{hom}(x′_{α}; \overline{ξ}),

and so

I_{\{ε_{n}\}}(v; Q′(x_{α}^{0}, δ)) ⩾ lim sup_{h→+∞} ∑_{i=1}^{h^{2}} \frac{h^{2}}{δ^{2}} ∫_{Q′_{i,h}} \frac{2δ^{2}}{h^{2}} W_{hom}(x′_{α}; \overline{ξ})dx′_{α} = 2 ∫_{Q′(x_{α}^{0},δ)} W_{hom}(x′_{α}; \overline{ξ})dx′_{α}.

Dividing both sides of the previous inequality by δ^{2} and passing to the limit when δ ↘ 0^{+}, we obtain by (3.4) and (3.6)

W_{\{ε_{n}\}}(x_{α}^{0}; \overline{ξ}) ⩾ W_{hom}(x_{α}^{0}; \overline{ξ}).
■

Proposition 3.6. W_{\{ε_{n}\}}(x_{α}; \overline{ξ}) = W_{hom}(x_{α}; \overline{ξ}) a.e. x_{α} ∈ ω and all \overline{ξ} ∈ ℝ^{3×2}.

Proof. Let E be the intersection of the set L (see Definition 3.3) with the subset of points x_{α}^{0} ∈ ω where W_{{ε_{n}}}(x_{α}^{0}; ·) and W_{hom}(x_{α}^{0}; ·) are continuous (see Lemma 2.7). Then L^{2}(ω \ E) = 0 and in view of Lemma 3.4 and 3.5, we have that for all x_{α}^{0} ∈ E and for all \overline{ξ} ∈ ℚ^{3×2}, W_{\{ε_{n}\}}(x_{α}^{0}; \overline{ξ}) = W_{hom}(x_{α}^{0}; \overline{ξ}). Since W_{\{ε_{n}\}}(x_{α}^{0}; ·) and W_{hom}(x_{α}^{0}; ·) are continuous for all x_{α}^{0} ∈ E, the equality W_{\{ε_{n}\}}(x_{α}^{0}; \overline{ξ}) = W_{hom}(x_{α}^{0}; \overline{ξ}) holds true for x_{α}^{0} ∈ E and all \overline{ξ} ∈ ℝ^{3×2}.

■

Corollary 3.7. Γ(L^{p}(A × I))- lim_{ε} I_{ε}( ·; A) = I_{hom}( ·; A) for all A ∈ A(ω), where I_{hom}(·; A) is the functional defined in (3.1).

Proof. From Proposition 3.6 we can conclude that I_{hom}(·; A) is well defined and

Γ(L^{p}(A × I))- lim_{n} I_{ε_{n}}( ·; A) = I_{hom}( ·; A)

for all A ∈ A(ω) (see Remark 3.1). Since this limit does not depend upon the extracted subsequence, in view of Proposition 7.11 in Braides and Defranceschi [9], the whole sequence {I_{ε}( ·; A)}_{ε>0} Γ(L^{p}(A × I))-converges to I_{hom}( ·; A) for each A ∈ A(ω).

■

The proof of Theorem 1.2 comes as a consequence of Corollary 3.7 taking A = ω.

4 Appendix

We now prove a technical result of extension of Carathéodory functions that was useful in the proof of Lemma 3.5. The argument used is very close to that of Theorem 1, Section 1.2 in Evans and Gariepy [15].

Lemma 4.1. Let Ω ⊂ ℝ^{N} be a bounded open set and f : Ω × ℝ^{m} × ℝ^{d×N} → ℝ a function such that

f(x, ·; ·) is continuous for a.e. x ∈ Ω;

f( · y; ξ) is L^{N}-measurable for all y ∈ ℝ^{N} and ξ ∈ ℝ^{N};

f(x, ·; ξ) is (0, 1)^{m}-periodic for a.e. x ∈ Ω and all ξ ∈ ℝ^{d×N}.

Assume also that there exists β > 0 and 1 ⩽ p < ∞ such that

\frac{1}{β} |ξ|^{p} − β ⩽ f(x, y; ξ) ⩽ β(1 + |ξ|^{p}), for a.e. x ∈ Ω and all (y, ξ) ∈ ℝ^{m} × ℝ^{d×N}.

Then for any η > 0 and λ > 0 there exist a compact set K_{η} ⊂ Ω and a function f^{η,λ} : ℝ^{N} ×ℝ^{m} ×ℝ^{d×N} → ℝ such that

L^{N}(Ω \ K_{η}) < η,
f^{η,λ}(x, y; ξ) = f(x, y; ξ) for all (x, y; ξ) ∈ K_{η} × ℝ^{m} × \overline{B}(0, λ),
f^{η,λ}( · , y; ·) is continuous for all y ∈ ℝ^{m},
f^{η,λ}(x, ·; ξ) is continuous and (0, 1)^{m}-periodic for all (x, ξ) ∈ ℝ^{N} × ℝ^{d×N},

and

−β ⩽ f^{η,λ}(x, y; ξ) ⩽ β(1 + λ^{p}), for all (x, y, ξ) ∈ ℝ^{N} × ℝ^{m} × ℝ^{d×N}.  (4.1)

Proof. Since f is a Carathéodory function, by Scorza Dragoni’s Theorem (see Ekeland and Teman [14]) for all η > 0 there exists a compact set K_{η} ⊂ Ω satisfying L^{N}(Ω \ K_{η}) < η and such that f is continuous on K_{η} × ℝ^{m} × ℝ^{d×N}. Let C^{η,λ} := K_{η} × \overline{B}(0, λ) ≡ C (to simplify notation) and U^{η,λ} := (ℝ^{N} × ℝ^{d×N}) \ C^{η,λ} ≡ U. Fix (s, F) ∈ C, and for all (x, ξ) ∈ U set

u_{(s,F)}^{η,λ}(x, ξ) := max \{2 − \frac{|(s, F) − (x, ξ)|}{dist((x, ξ), C)}, 0\} ≡ u_{(s,F)}(x, ξ).

Clearly

u_{(s,F)} is continuous on U,
0 ⩽ u_{(s,F)} ⩽ 1,
u_{(s,F)}(x, ξ) = 0 if and only if |(s, F) − (x, ξ)| ⩾ 2dist((x, ξ), C).

Let \{s_{j}^{η}\}_{j⩾1} ≡ \{s_{j}\}_{j⩾1} and \{F_{j}^{λ}\}_{j⩾1} ≡ \{F_{j}\}_{j⩾1} be a countable dense family of subsets of K_{η} and \overline{B}(0, λ), respectively. Define

σ^{η,λ}(x, ξ) := ∑_{j⩾1} 2^{−j}u_{(s_{j},F_{j})}(x, ξ) ≡ σ(x, ξ) for all (x, ξ) ∈ U.

Since σ is the uniform limit of a sequence of continuous functions in U, then σ is continuous in U. Moreover, for all (x, ξ) ∈ U, 0 < σ(x, ξ) ⩽ 1. Indeed, assume that σ(x, ξ) = 0 for some (x, ξ) ∈ U. Then, for all j ⩾ 1, u_{(s_{j},F_{j})}(x, ξ) = 0 and thus |(s_{j}, F_{j}) − (x, ξ)| ⩾ 2 dist((x, ξ), C). The density of {s_{j}, F_{j}} in C yields that |(s, F) − (x, ξ)| ⩾ 2 dist((x, ξ), C) for all (s, F) ∈ C. We obtain a contradiction if we choose (s, F) to be those points of C such that dist((x, ξ), C) = dist((x, ξ), (s, F)) so σ(x, ξ) > 0 for all (x, ξ) ∈ U. Consequently, the function

(x, ξ) ↦ v_{k}(x, ξ) ≡ v_{k}^{η,λ}(x, ξ) := \frac{2^{−k}u_{(s_{k},F_{k})}(x, ξ)}{σ(x, ξ)}

is well defined and continuous in U. Moreover it satisfies that

0 ⩽ v_{k}(x, ξ) ⩽ 1, ∑_{k⩾1} v_{k}(x, ξ) = 1 for all (x, ξ) ∈ U.

Fix y ∈ ℝ^{m} and define the continuous extension of f(·, y; ·) outside C as

f^{η,λ}(x, y; ξ) = \{ \begin{matrix}f(x, y, ξ) & if (x, ξ) ∈ C, \\ ∑_{k⩾1} v_{k}(x; ξ) f(s_{k}, y; F_{k}) & if (x, ξ) ∈ U. \end{matrix}

Obviously, we have f^{η,λ}(x, y; ξ) = f(x, y; ξ) for all (x, y; ξ) ∈ K_{η} × ℝ^{m} × \overline{B}(0, λ). On the other hand, if (x, y, ξ) is such that (x, ξ) ∈ U, in view of the p-growth and the p-coercivity condition on f we get that

−β ⩽ f^{η,λ}(x, y; ξ) ⩽ ∑_{k⩾1} v_{k}(x; ξ)β(1 + |F_{k}|^{p}) ⩽ β(1 + λ^{p}).

Since we have that

sup_{y∈ℝ^{m}, (x,ξ)∈U} [ ∑_{k⩾n} |2^{−k}u_{(s_{k},F_{k})}(x, ξ)f(s_{k}, y; F_{k})| ] ⩽ β(1 + λ^{p}) ∑_{k⩾n} 2^{−k} ⟶_{n→+∞} 0,  (4.2)

then the function

(x, y; ξ) ↦ ∑_{k⩾1} 2^{−k}u_{(s_{k},F_{k})}(x, ξ)f(s_{k}, y; F_{k})

is continuous on {(x, y, ξ) : (x, ξ) ∈ U, y ∈ ℝ^{m}}. In particular, for all (x, ξ) ∈ ℝ^{N} × ℝ^{d×N} the function f^{η,λ}(x, ·; ξ) is continuous. Further, f^{η,λ}(x, ·; ξ) it is (0, 1)^{m}-periodic because if i ∈ ℤ^{m} then for (x, ξ) ∈ U

f^{η,λ}(x, y + i; ξ) = ∑_{k⩾1} v_{k}(x; ξ) f(s_{k}, y + i; F_{k}) = ∑_{k⩾1} v_{k}(x; ξ) f(s_{k}, y; F_{k}) = f^{η,λ}(x, y; ξ).

Finally we prove the continuity of f^{η,λ}(·, y_{;}·). By (4.2) it suffices to show that for all (a, A) ∈ C

lim_{U∋(x,ξ)→(a,A)} f^{η,λ}(x, y; ξ) = f(a, y; A).

As {(s_{j}, F_{j})}_{j⩾1} is dense in C and f(·, y; ·) is continuous on C, for every ε > 0 there exists δ > 0 such that |f(a, y; A) − f(s_{j}, y; F_{j})| < ε for all j ⩾ 1 with |(a, A) − (s_{j}, F_{j})| < δ. Assume that |(x, ξ) − (a, A)| < δ/4 and suppose that j ⩾ 1 is such that |(a, A) − (s_{j}, F_{j})| ⩾ δ. Then

δ ⩽ |(a, A) − (s_{j}, F_{j})| ⩽ |(a, A) − (x, ξ)| + |(x, ξ) − (s_{j}, F_{j})| ⩽ \frac{δ}{4} + |(x, ξ) − (s_{j}, F_{j})|,

and thus

|(x, ξ) − (s_{j}, F_{j})| ⩾ \frac{3δ}{4} > 2|(a, A) − (x, ξ)| ⩾ 2 dist((x, ξ), C).

Consequently, v_{j}(x, ξ) = 0 if j is such that |(a, A) − (s_{j}, F_{j})| ⩾ δ, and so

|f^{η,λ}(x, y; ξ) − f(a, y; A)| ⩽ ∑_{j⩾1, |(a,A)−(s_{j},F_{j})|<δ} v_{j}(x, ξ)|f(s_{j}, y; F_{j}) − f(a, y; A)| < ε,

because non zero terms of the sum are those which satisfy |f(a, y; A) − f(s_{j}, y; F_{j})| < ε. The continuity of f^{η,λ}(·, y; ·) now follows.

■

Acknowledgements: The authors would like to thank Irene Fonseca and Gilles Francfort for suggesting them this research work, and for their fruitful comments and suggestions. The second author would like to thank Bernardo Sousa for his help with the graphic program. We wish to thank the Center for Nonlinear Analysis for its hospitality and support. The research of M. Baía was partially supported by Fundação para a Ciência e Tecnologia (Grant PRAXIS XXI SFRH\BD\1174\2000), Fundo Social Europeu, the Department of Mathematical Sciences of Carnegie Mellon University and its Center for Nonlinear Analysis.

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Jean-François Babadjian

L.P.M.T.M., Université Paris Nord, 93430, Villetaneuse, France E-mail address: jfb@galilee.univ-paris13.fr

Margarida Baía

Department of Mathematical Science, Carnegie Mellon University,

Pittsburgh, PA 15213, USA E-mail address: mbaia@andrew.cmu.edu