Multiscale Relaxation of Convex Functionals

Irene Fonseca Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.

Elvira Zappale Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, Università degli Studi di Salerno, Fisciano (Sa) 84084, Italy.

January 25, 2002

Abstract

The Γ-limit of a family of functionals

u ↦ ∫_{Ω} f ( \frac{x}{ε} , \frac{x}{ε^{2}} , D^{s}u) dx

is obtained for s = 1, 2 and when the integrand f = f(x, y, v) is a continuous function, periodic in x and y, and convex with respect to v. The 3-scale limits of second order derivatives are characterized.

Keywords: convexity, periodicity, multiscale limits, Γ-convergence, A-quasiconvexity, measurability selection criterion 2000 Mathematics Subject Classification: 35G99, 49J40, 49J45, 74G65

1 Introduction

The method of two-scale convergence was introduced by Allaire [A] and Nguetseng [N1] to render rigorous formal asymptotics techniques commonly used by practitioners in the theory of homogenization. Although it is well understood that periodicity poses severe constraints on physical realistic models, it is also agreed that the understanding of the effective behavior of periodically structured composite materials may shed light into the more complex and mathematically challenging media. For this reason, the theory of two-scale convergence, as well as its sequel, the theory of (n+1)-scale convergence developed by Allaire and Briane [ABr], have played a very important role in the theory of PDEs and its applications in homogenization (see also [N1]).

We may say that it all started with the result below, first established by Nguetseng in [N1], and subsequently generalized by Allaire and recasted with a new proof (see [A], Theorem 0.1).

Proposition 1.1. Let Ω be an open subset of ℝ^{N}, and let {u_{ε}} be a bounded sequence in L^{p}(Ω), 1 ≤ p < +∞. There exist a subsequence (not relabelled) and a function u_{0}(x, y) ∈ L^{p}(Ω × Q) such that

lim_{ε→0^{+}} ∫_{Ω} ψ (x, \frac{x}{ε}) u_{ε}(x) dx = ∫_{Ω} ∫_{Q} ψ(x, y) u_{0}(x, y) dx dy  (1.1)

for any smooth function ψ(x, y) which is Q−periodic in y. The (sub)sequence {u_{ε}} is said to two-scale converge to u_{0}(x, y).

Here, and in what follows, Q := (0, 1)^{N}, and Y_{i} are copies of Q for i = 1, …, n, n ∈ ℕ. In [A], and drawing from (1.1), Allaire introduced the concept of 2-scale convergence, which later he extended to (n+1)-scale convergence (n ∈ ℕ) in joint work with Briane [ABr].

Definition 1.2. A bounded sequence {u_{ε}} ⊂ L^{p}(Ω) is said to (n + 1)−scale converge to a function u_{0} ∈ L^{p} (Ω × Y_{1} × … × Y_{n}), 1 ≤ p < +∞, if

lim_{ε→0^{+}} ∫_{Ω} φ (x, \frac{x}{ε}, …, \frac{x}{ε^{n}} ) u_{ε}(x) dx = ∫_{Ω} ∫_{Y_{1}}⋯ ∫_{Y_{n}} φ(x, y_{1}, …, y_{n})u_{0}(x, y_{1}, …, y_{n}) dx dy_{1} … dy_{n}, (1.2)

for every φ ∈ L^{p′} (Ω; C_{per} (Y_{1} × … Y_{n})), p′ := \frac{p}{p−1}, and we write

u_{ε} ⇀^{(n+1)-s} u_{0}.

Remark 1.3. As shown in Corollary 5.4 in [A], admissible test functions φ may also be taken to be measurable in the oscillating variable and continuous with respect to x, precisely, writing φ̄(y, x) := φ(x, y) then φ̄ ∈ L_{per}^{p′}(Y_{1} × … Y_{n}; C(\overline{Ω})).

With the Definition 1.2 in hand, Allaire and Briane in [ABr] extended Proposition 1.1 to read

Proposition 1.4. Let Ω be an open subset of ℝ^{N}, and let {u_{ε}} be a bounded sequence in L^{p}(Ω), 1 ≤ p < +∞. There exist a subsequence (not relabelled) and a function u_{0} ∈ L^{p} (Ω × Y_{1} × … × Y_{n}) such that {u_{ε}} (n + 1)−scale converges to u_{0}.

Remark 1.5. The original result in [ABr] was stated for L^{2} but it can be generalized easily to L^{p}, 1 ≤ p < +∞. Note that if the sequence {u_{ε}} is bounded in L^{∞}(Ω), then u_{0} ∈ L^{∞}(Ω × Y_{1} × … × Y_{n}). Indeed, extracting a further subsequence (not relabelled) such that, for all k ∈ ℕ, {u_{ε}} (n + 1)−scale converges to u_{k} in L^{p}(Ω × Y_{1} × … × Y_{n}), it follows that u_{k} = u_{0} for all k and (see [A], Proposition 1.6)

∥u_{0}∥_{L^{∞}(Ω×Y_{1}×...×Y_{n})} = lim_{k→∞} ∥u_{0}∥_{L^{p}(Ω×Y_{1}×...×Y)} ≤ lim inf_{k→∞} lim sup_{ε→0^{+}} ∥u_{ε}∥_{L^{p}(Ω)} ≤ sup_{ε>0} ∥u_{ε}∥_{L^{∞}(Ω)} < +∞.

For this reason, a bounded sequence {u_{ε}} ⊂ L^{∞}(Ω) is said to (n + 1)-scale converge to u_{0} ∈ L^{∞}(Ω × Y_{1} × … × Y_{n}) if (1.2) holds for every φ ∈ L^{1} (Ω; C_{per} (Y_{1} × … Y_{n})).

A class of functions for which it is easy to identify the (n + 1)-scale limit is that of constant functions. We start by recalling the well known Riemann-Lebesgue Lemma.

Lemma 1.6. Let w ∈ L_{per}^{p}(Q; ℝ^{d}), 1 ≤ p < +∞, and let w_{ε}(x) := w (\frac{x}{ε}), ε > 0. If E ⊂ ℝ^{N} is a measurable set then

w_{ε}⇀ ∫_{Q} w(y) dy in L_{loc}^{p}(E, ℝ^{d}) (⇀^{⋆} if p = ∞).  (1.3)

Donato [Do] (see also [AHZ]) has extended (1.3) to the realm of multiscale convergence by showing that

∫_{Ω} φ (x, \frac{x}{ε}, … \frac{x}{ε^{n}} ) dx → ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{n}} φ(x, y_{1}, …, y_{n}) dx dy_{1} … dy_{n}  (1.4)

whenever φ ∈ L^{p′}(Ω; C_{per}(Y_{1} × ⋯ × Y_{n})). In particular, a sequence identically constant (n + 1)−scale converges to the constant itself.

Also, Allaire and Briane have characterized fully the (n+1)-scale limits of gradients in the theorem below (see [ABr], Thm 2.6).

Theorem 1.7. For any bounded sequence {u_{ε}} ⊂ W^{1,p}(Ω) there exist u ∈ W^{1,p}(Ω), u_{1} ∈ L^{p}(Ω; W_{per}^{1,p}(Y_{1})) and n functions u_{k} ∈ L^{p}(Ω×Y_{1}× … ×Y_{n−1}; W_{per}^{1,p}(Y_{n})), k = 1, …, n, such that, up to a subsequence,

u_{ε} ⇀^{(n+1)-s} u,  (1.5)

Du_{ε} ⇀^{(n+1)-s} Du(x) + ∑_{k=1}^{n} D_{y_{k}}u_{k}(x, y_{1}, …, y_{n}).  (1.6)

Moreover, any (n + 1)−tuple limit (u, u_{1}, …, u_{n}) ∈ W^{1,p}(Ω) × L^{p}(Ω; W_{per}^{1,p}(Y_{1})) × ⋯ × ×L^{p}(Ω×Y_{1}× … ×Y_{n−1}; W_{per}^{1,p}(Y_{n})) may be (n + 1)−scale attained, i.e. there exists a bounded sequence {u_{ε}} in W^{1,p}(Ω; ℝ^{d}) for which (1.5) and (1.6) hold.

Using the notion of 2-scale convergence Allaire provided a simple and elegant proof for the “homogenized” Γ-limit of a functional (see [A] Theorems 3.1 and 3.3; see also [Ma1])

I_{ε}(u) := ∫_{Ω} W (\frac{x}{ε}, Du) dx.

Precisely,

Theorem 1.8. Let W : ℝ^{N} × ℝ^{d×N} → [0, +∞) be an integrand satisfying

(i) W (·, ξ) is measurable and Q-periodic for all ξ ∈ ℝ^{d×N};

(ii) W (x, ·) is convex and C^{1} for a.e. x ∈ ℝ^{N};

(iii) \frac{1}{C} |ξ|^{p} ≤ W (x, ξ) ≤ C(1 + |ξ|^{p}) for some p > 1, C > 0, for all ξ ∈ ℝ^{N} and for a.e. x ∈ ℝ^{N}.

Then

Γ(L^{p}(Ω)) − lim_{ε→+0} I_{ε}(u) = ∫_{Ω} W_{hom}(Du)dx

for every u ∈ W^{1,p}(Ω; ℝ^{d}), where

W_{hom}(ξ) := inf {∫_{Q} W (y, ξ + Dφ(y)) dy : φ ∈ W_{per}^{1,p}(Q; ℝ^{d})} .

Note that in Theorems 3.1 and 3.3 in [A] Allaire assumed further a growth condition on \frac{∂W}{∂ξ} (x, ξ) (see (4.8)), but this latter hypothesis is superfluous in view (iii) and the convexity of W (x, ·).

It is our aim in this paper to extend Theorem 1.8 to multiscale, higher order, convex, and periodic variational problems. As before, let Ω be an open, bounded subset of ℝ^{N}, let s ∈ {1, 2} and consider a function

f : ℝ^{N} × ℝ^{N} × ℝ_{∗}^{s} → [0, +∞)

where ℝ_{∗}^{1} := ℝ^{d×N} and ℝ_{∗}^{2} := (Sym(ℝ^{N}, ℝ^{N}))^{d}, with Sym(ℝ^{N}, ℝ^{N}) being the space of all linear symmetric transformations from ℝ^{N} onto ℝ^{N}.

We seek to characterize the asymptotic behavior as ε→0^{+} of functionals

u ∈ W^{s,p}(Ω; ℝ^{d}) ↦ ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , D^{s}u(x)) dx

by providing an integral representation for the effective energy.

Our main result is the following:

Theorem 1.9. Let f : ℝ^{N} × ℝ^{N} × ℝ_{∗}^{s} → [0, +∞) be an integrand satisfying the assumptions:

(H1) f is continuous in ℝ^{N} × ℝ^{N} × ℝ_{∗}^{s};

(H2) f is separately Q−periodic in x and y;

(H3) f (x, y, ·) is convex for all x, y ∈ ℝ^{N};

(H4) \frac{1}{C} |ξ|^{p} ≤ f(x, y, ξ) ≤ C(1 + |ξ|^{p}) for some p > 1, C > 0 and for all (x, y, ξ) ∈ ℝ^{N} × ℝ^{N} × ℝ_{∗}^{s}.

Then

Γ(L^{p}(Ω)) − lim_{ε→0^{+}} J_{ε}(u, Ω) = ∫_{Ω} \overline{f_{hom}^{s}}(D^{s}u)dx  (1.7)

for every u ∈ W^{s,p}(Ω; ℝ^{d}), where

\overline{f_{hom}^{s}}(ξ) := inf \{∫_{Q} f_{hom}^{s}(x, ξ + D^{s}φ(x)) dx : φ ∈ W_{per}^{s,p}(Q; ℝ^{d})\}  (1.8)

and

f_{hom}^{s}(x, ξ) = inf {∫_{Q} f(x, y, ξ + D^{s}ψ(y)) dy : ψ ∈ W_{per}^{s,p}(Q; ℝ^{d})} .  (1.9)

Remark 1.10. (i) It is not clear what is the natural regularity that f = f(x, y, ξ) should satisfy with respect to x and y. A thorough discussion of the difficulties encountered when only measurability is required on the pair (x, y) may be found in Allaire [A]. Indeed, in this case not even the measurability of, say, f(x/ε, x/ε^{2}, 0) is guaranteed, letting alone the validity of results such as (1.4) (see also [Do2]). As Allaire points out, this measurability is ensured when there is continuity on least in one of the arguments, and here we choose it to be x (see (A1), (A2) below). (ii) Theorem 1.9 was proven in the case where s = 1 by Braides and Lukkassen (see [BL], Theorem 1.1; see also [BrD]), where (H1) was on one hand relaxed to read

(H1)′ f (·, y, ·) is continuous for a.e. y ∈ ℝ^{N} and f(x, ·, ξ) is measurable for all (x, ξ) ∈ Ω × ℝ^{d×N},

and on the other hand it was strengthened with the additional uniform continuity assumption

(H5)|f(x, y, ξ)−f(x′, y, ξ)| ≤ ω(|x−x′|)(a(y)+f(x, y, ξ)) for all x, x′ ∈ ℝ^{N}, a.e. y ∈ ℝ^{N}, for all ξ ∈ ℝ^{d×N}, where ω is a continuous, positive function with ω(0) = 0.

Note that this latter condition essentially entails some sort of decoupling between different scales. (iii) As it can be seen directly from the proof of Theorem 1.9, by analogy with the statement of Theorem 1.8, it is possible to replace condition H1) with the assumptions

(A1) f (·, y, ·) is continuous for a.e. y ∈ ℝ^{N};

(A2) f(x, ·, ξ) is measurable for all (x, ξ) ∈ Ω × ℝ_{∗}^{s};

(A3) \frac{∂f}{∂ξ} (x, y, ξ) exists for all x ∈ Ω, a.e. y ∈ ℝ^{N}, and for all ξ ∈ ℝ^{d×N}, and it satisfies (A1), (A2).

(iv) Theorem 1.9 has also been extended to the non convex case and for s = 1 by Braides and Defranceschi [BrD] under assumptions (H1)′ , (H2), (H4) and (H5), and by Fonseca and Leoni [FL] under the hypotheses (H1), (H2) and (H4). The latter approach relies entirely on the blow-up method.

The main idea of the proof of Theorem 1.9 is to combine a three-scale convergence generalization of Allaire’s 2-scale argument with iterated integrals to obtain the lower bound in (1.7) (see [A] Theorem 3.1 and 3.3; see also [MT]), with the blow-up method introduced in [FM1] to assert the upper bound. To this end, we extend Theorem 1.7 to second order derivatives to read

Theorem 1.11. If {u_{ε}} is a bounded sequence in W^{2,p}(Ω; ℝ^{d}), ε > 0, then there exists a subsequence (not relabelled) converging weakly in W^{2,p}(Ω; ℝ^{d}) to a function u, and there exist U ∈ L^{p}(Ω; W^{2,p}(Q; ℝ^{d})) and W ∈ L^{p}(Ω×Q; W^{2,p}(Q; ℝ^{d})) such that

(i) U − A(x)y ∈ L^{p}(Ω; W_{per}^{2,p}(Q; ℝ^{d})) for some A ∈ L^{p}(Ω; ℝ^{d×N});

(ii) W − C(x, y)z ∈ L^{p}(Ω×Q; W_{per}^{2,p}(Q; ℝ^{d})) for some C ∈ L^{p}(Ω×Q; ℝ^{d×N});

(iii) u_{ε} ⇀^{3-s} u, Du_{ε} ⇀^{3-s} Du, and

\frac{∂^{2}u_{ε}}{∂x_{i}∂x_{j}} ⇀^{3-s} \frac{∂^{2}u}{∂x_{i}∂x_{j}} + \frac{∂^{2}U}{∂y_{i}∂y_{j}} (x, y) + \frac{∂^{2}W}{∂z_{i}∂z_{j}} (x, y, z).  (1.10)

Conversely, given u ∈ W^{2,p}(Ω; ℝ^{d}), U ∈ L^{p}(Ω; W^{2,p}(Q; ℝ^{d})), and W ∈ L^{p}(Ω×Q; W^{2,p}(Q; ℝ^{d})) satisfying (i), (ii), there exists a bounded sequence {u_{ε}} ⊂ W^{2,p}(Ω; ℝ^{d}) for which (iii) holds.

2 Preliminaries

In what follows Ω is an open, bounded domain in ℝ^{N}, Q := (0, 1)^{N}, Y_{1}, …, Y_{n}, are n identical copies of Q, with n ∈ ℕ, Q(x, ε) := x + εQ for x ∈ ℝ^{N} and ε > 0, ℝ_{∗}^{1} := ℝ^{d} and ℝ_{∗}^{2} := (Sym(ℝ^{N}, ℝ^{N}))^{d}, where Sym(ℝ^{N}, ℝ^{N}) is the space of all linear symmetric transformations from ℝ^{N} onto ℝ^{N}. C_{c}^{k}(ℝ^{N}; ℝ^{d}) is the space of k-differentiable ℝ^{d}-valued functions in Ω with compact support, and C_{per}^{k}(ℝ^{N}; ℝ^{d}) stands for the space of Q-periodic functions in C^{k}(ℝ^{N}; ℝ^{d}), 1 ≤ k ≤ +∞. Recall that f is said to be Q-periodic if f(x + ke_{i}) = f(x) for all x, all k ∈ ℤ^{k}, and for all i = 1, …, N, where {e_{1}, …, e_{N}} is the standard orthonormal basis of ℝ^{N}. C_{c}(ℝ^{N}) stands for the space of continuous functions with compact support in ℝ^{N}, C_{per}(ℝ^{N}) is the space of periodic continuous functions on ℝ^{N}, L_{per}^{p}(ℝ^{N}; ℝ^{d}) is the closure in L_{loc}^{p}(ℝ^{N}; ℝ^{d}) of C_{per}^{∞}(ℝ^{N}; ℝ^{d}), and W_{per}^{k,p}(Ω; ℝ^{d}) is the closure in W_{loc}^{k,p}(ℝ^{N}; ℝ^{d}) of C_{per}^{∞}(ℝ^{N}; ℝ^{d}). L^{N} is the N−dimensional Lebesgue measure, and A(Ω) is the family of open subsets of Ω. C will stand for a generic positive constant which may vary from expression to expression within the same formula.

The following well known lowersemicontinuity result will be used in the proof of the continuity of f_{hom} (for a proof we refer the reader to [I1], [I2]).

Theorem 2.1. Let Ω be an open, bounded subset of ℝ^{N}, and let f : Ω × ℝ^{d} × ℝ^{m} → [0, +∞), f = f(x, u, v), be a function Lebesgue measurable with respect to x, and Borel measurable with respect to (u, v). Suppose further that f(x, ·, ·) is lower semicontinuous for a.e. x ∈ Ω, f(x, u, ·) is convex for a.e. x ∈ Ω, u ∈ ℝ^{d}, and there exists (u_{0}, v_{0}) ∈ L^{q}(Ω; ℝ^{d}) × L^{p}(Ω; ℝ^{m}) with 1 ≤ p, q < +∞ such that

∫_{Ω} f(x, u_{0}(x), v_{0}(x)) dx < +∞.

If u_{n} → u in L^{q}(Ω; ℝ^{d}) and v_{n} ⇀ v in L^{p}(Ω; ℝ^{m}) then

∫_{Ω} f(x, u, v) dx ≤ ∫_{Ω} lim inf_{n→∞} f(x, u_{n}, v_{n}) dx.

Next we recall De Giorgi’s notion of Γ–convergence. Let (X, d) be a metric space and let {F_{ε}} be a family of functionals defined on (X, d). We say that a functional F : X × A(Ω)→[0, +∞] is the Γ(X)–lim inf (resp. the Γ(X)–lim sup) of the family {F_{ε}} if for every sequence {ε_{n}} converging to 0^{+}

F(u, Ω) = inf {lim inf_{n→+∞} ( resp. lim sup_{n→+∞})F_{ε_{n}}(u_{n}, Ω) : u_{n}→u in (X, d)} ,

and we write

F = Γ(X) − lim inf_{ε→0^{+}} F_{ε} (resp. F = Γ(X) − lim sup_{ε→0^{+}} F_{ε}).

F is the Γ(X)-limit of the family {F_{ε}}, and we write

F = Γ(X) − lim_{ε} F_{ε},

if Γ(X) − lim inf and Γ(X) − lim sup coincide.

The following result is well known result in the theory of Γ-convergence, and for the convenience of the reader its proof may be found in the Appendix (see also [BhFF], [BrFF]).

Proposition 2.2. Let F_{ε} : W^{1,p}(Ω; ℝ^{m})×A(Ω) → [0, +∞), ε > 0, be a sequence of functionals satisfying the hypotheses

(i) F_{ε}(u, ·) is the restriction to A(Ω) of a Radon measure;

(ii) F_{ε}(u, A) = F_{ε}(v, A) whenever u = v a.e. in A ∈ A(Ω);

(iii) there exists a positive constant C such that for all A ∈ A(Ω)

\frac{1}{C} (∫_{A} |Du|^{p} dx) ≤ F_{ε}(u, A) ≤ C ∫_{A} (1 + |Du|^{p}) dx,  (2.1)

for every ε > 0.

For every sequence {ε_{n}} converging to 0^{+} there exists a subsequence {ε_{j}} such that the functional F_{{ε_{j}}} (u, A) is the Γ(L^{p}(A)) limit of {F_{ε_{j}} (·, A)} for every A ∈ A(Ω) and u ∈ W^{1,p}(A; ℝ^{d}), where for any sequence {δ_{n}} converging to 0^{+}

F_{{δ_{n}}} (u, A) := inf {lim inf_{n→∞} F_{δ_{n}}(u, A) : u_{n} → u in L^{p}(A)} .

For a comprehensive study of Γ–convergence we refer to [Da] and [BrD].

The proof of Theorem 1.11 uses a version of Aumann’s Measurability Selection Principle and a simple corollary of the theory of A-quasiconvexity (see [D], [FM2], [BrFL]). We recall

Theorem 2.3. Let (X, M) be a measurable space with μ a positive, finite and complete measure, and let Z be a complete, separable metric space. Let F : X → {C ⊂ Z : C ≠ Ø, C is closed} be a multifunction such that {(x, y) ∈ X × Z : y ∈ F (x)} ∈ M × β(Z), where β(Z) is the Borel σ-algebra of Z. Then there exists a sequence of measurable functions f_{n} : X → Z such that

F (x) = \overline{\{f_{n}(x) : n ∈ ℕ\}} for μ a.e. x ∈ X.

As in [FM2], let

A : L^{q}(Ω; ℝ^{l}) → W^{−1,q}(Ω; ℝ^{m}), Av := ∑_{i=1}^{N} A^{(i)} \frac{∂v}{∂x_{i}} ,

be a constant–rank, first order linear partial differential operator, with A^{(i)} : ℝ^{l} → ℝ^{m} linear transformations, i = 1, …, N. We recall that A satisfies the constant-rank property if there exists r ∈ ℕ such that (see [Mu])

rank 𝔸w = r for all w ∈ S^{N−1},

where

𝔸w := ∑_{i=1}^{N} w_{i} A^{(i)}, w ∈ ℝ^{N}.

For each w ∈ ℝ^{N} the operator ℙ (w) : ℝ^{l} → ℝ^{l} is the orthogonal projection of ℝ^{l} onto ker 𝔸 (w), and 𝕊 (w) : ℝ^{m} → ℝ^{l} is defined by 𝕊 (w) 𝔸 (w) z := z − ℙ (w) z for z ∈ ℝ^{l} and 𝕊 ≡ 0 on (range(𝔸(w))^{⊥}. It may be shown that ℙ : ℝ^{N}\ {0} →Lin(ℝ^{l}; ℝ^{l}) is smooth and homogeneous of degree zero and 𝕊 : ℝ^{N}\ {0} →Lin(ℝ^{m}; ℝ^{l}) is smooth and homogeneous of degree −1.

We introduce the operators

S : L_{per}^{p} (Q; ℝ^{m}) → W_{per}^{1,p} (Q; ℝ^{l}) , T : L_{per}^{p} (Q; ℝ^{l}) → L_{per}^{p} (Q; ℝ^{l})

given by

Sv (x) := ∑_{λ∈ℤ^{N} \{0}} 𝕊 (λ) v̂ (λ) e^{2πix·λ}

and

T v (x) := ∑_{λ∈ℤ^{N}} v̂ (λ) e^{2πix·λ} − ∑_{λ∈ℤ^{N} \{0}} 𝕊 (λ) 𝔸 (λ) v̂ (λ) e^{2πix·λ}  (2.2)

whenever

v (x) = ∑_{λ∈ℤ^{N}} v̂ (λ) e^{2πix·λ},

and where the Fourier coefficients of a function f ∈ L_{per}^{1} (Q) are defined by

f̂(λ) := ∫_{Q} f (x) e^{−2πix·λ}dx, λ ∈ ℤ^{N}.

The following proposition may be found in [FM2].

Proposition 2.4. T : L_{per}^{p} (Q; ℝ^{l}) → L_{per}^{p} (Q; ℝ^{l}) is a bounded linear operator that vanishes on constant mappings and S : L_{per}^{p} (Q; ℝ^{m}) → W_{per}^{1,p} (Q; ℝ^{l}) is a pseudo differential bounded operator of order −1 such that (i) if v ∈ L_{per}^{p} (Q; ℝ^{l}) then T ∘T v = T v and A (T v) = 0;

(ii) ||v − T v||_{L_{per}^{p}(Q;ℝ^{l})} ≤ C_{p}||A (v) ||_{W_{per}^{−1,p}(Q;ℝ^{m})} for all v ∈ L_{per}^{p} (Q; ℝ^{l}) such that ∫_{Q} v dx = 0, for some C_{p} > 0;

(iii) v − T v = SAv for all v ∈ L_{per}^{p} (Q; ℝ^{l}).

Just as in the case of first order gradients (see [K], [FMP]), L^{p} bounded, A-constrained sequences may be modified on small sets so as to render them equi-integrable. Precisely (see [FM2]),

Proposition 2.5. Let 1 < p < +∞, let {V_{n}} be a bounded sequence in L^{p}(Ω; ℝ^{l}) such that AV_{n} → 0 in W^{−1,p}(Ω; ℝ^{l}), V_{n} ⇀ V in L^{p}(Ω; ℝ^{l}). Then there exists a sequence {v_{n}} ⊂ L^{p}(Ω; ℝ^{l}) ∩ ker(A) such that {|v_{n}|^{p}} is equi-integrable,

lim_{n→∞} L^{N} ({x ∈ Ω : V_{n}(x) ≠ v_{n}(x)}) = 0,  (2.3)

and

∫_{Ω} v_{n} dx = ∫_{Ω} V dx, ||v_{n} − V_{n}||_{L^{s}(Ω)} → 0 for all 1 ≤ s < p.

Moreover, if Ω = Q then v_{n} − V ∈ L_{per}^{p}(ℝ^{N}; ℝ^{l}) ∩ ker(A).

The relevance of this general framework, as emphasized by Tartar (see [T1], [T2], [T3], [T4], [T5]), lies on the fact that in continuum mechanics and electromagnetism PDEs other than curl v = 0 arise naturally, and this calls for a relaxation theory which encompasses PDE constraints of the type Av = 0. Included in this general setting are gradients of arbitrary order. Indeed, for gradients of order zero (i.e., unconstrained fields) it suffices to set Av ≡ 0. Here, due to Jensen’s inequality A-quasiconvexity reduces to convexity. For gradients of order one, Av = 0 if and only if curl v = 0. Note that w ∈ C_{per}^{∞}(ℝ^{N}; ℝ^{l}) is such that curl w = 0 and ∫_{Q} w(y) dy = 0 if and only if there exists φ ∈ C_{per}^{∞}(ℝ^{N}; ℝ^{d}) such that ∇φ = v, where l = d × N. Thus, in this case we recover the well known notion of quasiconvexity introduced by [Mo] (see also [D]). For higher order gradients we replace the target space ℝ^{l} by an appropriate finite dimensional vector space E_{s}^{d} of d-tuples of symmetric s-linear maps on ℝ^{N}, and it is then possible to find a first order linear partial differential operator A such that v ∈ L^{p}(Ω; E_{s}^{d}) and Av = 0 if and only if there exists φ ∈ W^{s,p}(Ω; ℝ^{d}) such that v = ∇^{s}φ.

3 Proof of Theorem 1.11

We are now in position to characterize the multiscale limit for the Hessian tensor. In order to simplify the exposition, we consider the case where n = 2, although the proof may be carried out for arbitrary n ∈ ℕ with the obvious adaptations.

Proof of Theorem 1.11. By working with each coordinate separately, without loss of generality we may assume that u_{ε} is scalar valued. Applying Theorem 1.7 to the sequence {(u_{ε}, Du_{ε})} ⊂ W^{1,p}(Ω; ℝ×ℝ^{N}), we get

u_{ε}(x) ⇀^{3-s} u(x), Du_{ε}(x) ⇀^{3-s} Du(x)

and

\frac{∂^{2}u_{ε}}{∂x_{i}∂x_{j}} = \frac{∂}{∂x_{i}} (\frac{∂u_{ε}}{∂x_{j}} ) ⇀^{3-s} \frac{∂^{2}u}{∂x_{i}∂x_{j}} (x) + \frac{∂U^{(j)}}{∂y_{i}} (x, y) + \frac{∂W^{(j)}}{∂z_{i}} (x, y, z),  (3.1)

for some U^{(j)} ∈ L^{p}(Ω; W_{per}^{1,p}(Y_{1})) and W^{(j)} ∈ L^{p}(Ω×Y_{1}; W_{per}^{1,p}(Y_{2})), j=1, …, N.

For any function φ ∈ C_{c}^{∞}(Ω; C_{per}^{∞}(Y_{1}×Y_{2})) we have

∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} (\frac{∂U^{(j)}}{∂y_{i}} (x, y) − \frac{∂U^{(i)}}{∂y_{j}} (x, y) + \frac{∂W^{(j)}}{∂z_{i}} (x, y, z) − \frac{∂W^{(i)}}{∂z_{j}} (x, y, z)) φ(x, y, z) dx dy dz = 0,

where we have used the fact that \frac{∂^{2}u_{ε}}{∂x_{i}∂x_{j}} = \frac{∂^{2}u_{ε}}{∂x_{j}∂x_{i}}. Choosing φ(x, y, z) := θ(x)ψ(y, z) with θ ∈ C_{c}^{∞}(Ω), ψ ∈ C_{per}^{∞}(Y_{1} × Y_{2}), we get for a.e. x ∈ Ω

∫_{Y_{1}} ∫_{Y_{2}} (\frac{∂U^{(j)}}{∂y_{i}} (x, y) − \frac{∂U^{(i)}}{∂y_{j}} (x, y) + \frac{∂W^{(j)}}{∂z_{i}} (x, y, z) − \frac{∂W^{(i)})}{∂z_{j}}(x, y, z)) ψ(y, z)dydz = 0.  (3.2)

If, in addition, ψ(y, z) = ψ(y) then (3.2) becomes

0 = ∫_{Y_{1}} (\frac{∂U^{(j)}}{∂y_{i}} (x, y) − \frac{∂U^{(i)}}{∂y_{j}} (x, y)) ψ(y) dy + ∫_{Y_{1}} ψ(y) (∫_{Y_{2}} (\frac{∂W^{(i)}}{∂z_{j}} (x, y, z) − \frac{∂W^{(j)}}{∂z_{i}} (x, y, z)) dz) dy

= ∫_{Y_{1}} (\frac{∂U^{(j)}}{∂y_{i}} (x, y) − \frac{∂U^{(i)}}{∂y_{j}} (x, y)) ψ(y)dy

for a.e. x ∈ Ω, in light of the periodicity of W^{(i)}(x, y, ·) and W^{(j)}(x, y, ·). We conclude that

\frac{∂U^{(j)}}{∂y_{i}} (x, y) = \frac{∂U^{(i)}}{∂y_{j}} (x, y) for a.e. x ∈ Ω and y ∈ Y_{1}.  (3.3)

Now (3.2) reduces to

∫_{Y_{1}} ∫_{Y_{2}} (\frac{∂W^{(i)}}{∂z_{j}} − \frac{∂W^{(j)}}{∂z_{i}} ) ψ(y, z)dydz = 0,

and choosing as test function ψ(y, z) := ψ_{1}(y)ψ_{2}(z), with ψ_{1} and ψ_{2} smooth and periodic, we obtain

\frac{∂W^{(j)}}{∂z_{i}} (x, y, z) = \frac{∂W^{(i)}}{∂z_{j}} (x, y, z) for a.e. (x, y, z) ∈ Ω × Y_{1} × Y_{2}.  (3.4)

Note that in (3.2), (3.3) and (3.4) we have used the fact the tensor product space C_{per}^{∞}(X_{1})⊗ … ⊗C_{per}^{∞}(X_{n}) is dense in C_{per}^{∞}(X_{1}× … ×X_{n}), where X_{1}, … X_{n} are arbitrary Banach spaces.

We claim that there exists U ∈ L^{p}(Ω; W^{2,p}(Y_{1})) such that

\frac{∂U}{∂y_{i}} (x, y) = U^{(i)}(x, y)  (3.5)

for a.e. x ∈ Ω, y ∈ Y_{1}. We define

\overline{U} := (U^{(1)}, …, U^{(N)}) ∈ L^{p} (Ω; W_{per}^{1,p}(Y_{1})^{N}) .

Although it is clear from (3.3) that for a.e. x ∈ Ω there exists g_{x} ∈ W^{2,p}(Y_{1}) such that D_{y}g_{x}(y) = \overline{U}(x, y) a.e. y ∈ Y_{1}, we need to ensure the measurability and integrability in x of (x, y) ↦ g_{x}(y). Choose piecewise affine functions θ_{k} ∈ L^{p} (Ω; W_{per}^{1,p}(Y_{1})^{N}) such that

lim_{k→∞} ∥θ_{k} − \overline{U}∥_{L^{p}(Ω;W_{per}^{1,p}(Y_{1})^{N})} = 0.  (3.6)

Consider the first order linear system of PDEs given by Av := curl v for v ∈ L^{p} (Y_{1}; ℝ^{N}). We show that

∥A(θ_{k})∥_{L^{p}(Ω;L^{p}(Y_{1}))} → 0  (3.7)

where here A(θ_{k}) := curl_{y} θ_{k}(x, y) for a.e. x ∈ Ω. Indeed, for every i, j = 1, …, N, we have

∫_{Ω} ∥\frac{∂θ_{k}^{(i)}}{∂y_{j}} (x, ·) − \frac{∂θ_{k}^{(j)}}{∂y_{i}} (x, ·)∥_{L^{p}(Y_{1})}^{p} dx ≤ C ∫_{Ω} ∥\frac{∂θ_{k}^{(i)}}{∂y_{j}} (x, ·) − \frac{∂U^{(i)}}{∂y_{j}} (x, ·)∥_{L^{p}(Y_{1})}^{p} dx

+ C ∫_{Ω} ∥\frac{∂U^{(i)}}{∂y_{j}} (x, ·) − \frac{∂U^{(j)}}{∂y_{i}} (x, ·)∥_{L^{p}(Y_{1})}^{p} dx + C ∫_{Ω} ∥\frac{∂θ_{k}^{(j)}}{∂y_{i}} (x, ·) − \frac{∂U^{(j)}}{∂y_{i}} (x, ·)∥_{L^{p}(Y_{1})}^{p} dx

≤ C ∫_{Ω} ∥θ_{k}(x, ·) − \overline{U}(x, ·)∥_{W^{1,p}(Y_{1})}^{p} dx → 0, where we have used (3.3) and (3.6).

We are now in position to apply Theorem 2.3, where here (X, M) := (Ω, L), Z := W_{per}^{1,p}(Y_{1}), μ is the Lebesgue measure and L is the σ-algebra of Lebesgue measurable sets in ℝ^{N}. For each k ∈ ℕ consider

F_{k} : Ω → {C ⊂ W_{per}^{1,p}(Y_{1}) : C ≠ Ø, C is closed}
x ↦F_{k}(x) := {v ∈ W_{per}^{1,p}(Y_{1}) : ∫_{Y_{1}} v dy = 0, D_{y}v = T (θ_{k}(x, y))}

where T is the operator introduced in (2.2). Note that since T (θ_{k}) is curl free, then F_{k}(x) is nonempty for a.e. x ∈ Ω, and it is trivially closed. Note also that, with θ_{k}(x, y) = ∑_{i=1}^{k} χ_{A_{i}}(x)λ_{k}^{(i)}(y) for some k ∈ ℝ^{N}, A_{i} ⊂ Ω mutually disjoint and λ_{k}^{(i)} ∈ W_{per}^{1,p}(Y_{1}), we have

{(x, v) : x ∈ Ω, v ∈ F_{k}(x)} = ∪_{i=1}^{k}A_{i} × {v ∈ W^{1,p}(Y_{1}) : ∫_{Y_{1}} v dy = 0, D_{y}v = T (λ_{k}^{(i)})} ,

which belongs to L × β (W^{1,p}(Y_{1})) because the sets

{v ∈ W^{1,p}(Y_{1}) : ∫_{Y_{1}} vdy = 0, D_{y}v = T (λ_{k}^{(i)})}

are closed. By Theorem 2.3 we may find measurable functions f_{n,k} : Ω → W^{1,p}(Y_{1}) such that

F_{k}(x) = \overline{\{f_{n,k}(x)\}}.

Now set v_{k}(x) := f_{1,k}(x). By Poincaré-Friedrichs inequality, and in view of Proposition 2.4, we have

∫_{Ω} ∥v_{k}(x, ·)∥_{W_{per}^{1,p}(Y_{1})}^{p} dx ≤ C ∫_{Ω} ∥D_{y}v_{k}(x, ·)∥_{L_{per}^{p}(Y_{1})}^{p} dx
≤ C ∫_{Ω} ∥θ_{k}(x, ·)∥_{(L_{per}^{p}(Y_{1}))^{N}}^{p} dx → ∫_{Ω} ∥\overline{U}(x, ·)∥_{(L_{per}^{p}(Y_{1}))^{N}}^{p} dx,

and so there exists a subsequence (not relabelled), and there exists Ũ ∈ L^{p}(Ω; W_{per}^{1,p}(Y_{1})) such that

v_{k} ⇀ Ũ in L^{p}(Ω; W_{per}^{1,p}(Y_{1})).

Now since the norm is weakly lower semicontinuous, by Poincaré-Friedrichs inequality we have

∫_{Ω} ∥\frac{∂Ũ}{∂y_{i}} (x, ·) − (U^{(i)}(x, ·) − ∫_{Y_{1}} U^{(i)}(x, z) dz)∥_{L_{per}^{p}(Y_{1})}^{p} dx

≤ lim inf_{k→∞} ∫_{Ω} C ∥\frac{∂v_{k}}{∂y_{i}} (x, ·) − (U^{(i)}(x, ·) − ∫_{Y_{1}} U^{(i)}(x, z) dz)∥_{L_{per}^{p}(Y_{1})}^{p} dx

≤ lim sup_{k→∞} ∫_{Ω} C ∥\frac{∂v_{k}}{∂y_{i}} (x, ·) − (T (θ_{k}(x, ·)))^{(i)}∥_{L_{per}^{p}(Y_{1})}^{p} dx

+ lim sup_{k→∞} ∫_{Ω} C ∥(T (θ_{k}(x, ·)))^{(i)} − (θ_{k}^{(i)}(x, ·) − ∫_{Y_{1}} θ_{k}^{(i)}((x, z) dz)∥_{L_{per}^{p}(Y_{1})}^{p} dx

+ lim sup_{k→∞} ∫_{Ω} C ∥θ_{k}^{(i)}(x, ·) − ∫_{Y_{1}} θ_{k}^{(i)}(x, z) dz − (U^{(i)}(x, ·) − ∫_{Y_{1}} U^{(i)}(x, z) dz)∥_{L_{per}^{p}(Y_{1})}^{p} dx

=: I_{1} + I_{2} + I_{3},

where I_{1} = 0 by definition of F_{k}, (3.6) implies that I_{3} → 0, and by Proposition 2.4(ii) and (3.7) it follows that

I_{2} ≤ C lim sup_{k→∞} ∥A(θ_{k})∥_{L^{p}(Ω;W^{−1,p}(Y_{1}))} ≤ C lim sup_{k→∞} ∥A(θ_{k})∥_{L^{p}(Ω;L^{p}(Y_{1}))} → 0.

Therefore, setting A(x) := − ∫_{Y_{1}} \overline{U}(x, z) dz then A ∈ L^{p}(Ω; ℝ^{N}) and

\frac{∂U}{∂y_{i}} (x, y) = U^{(i)}(x, y)

for a.e. x ∈ Ω, y ∈ Y_{1}, where

U := Ũ + A(x) · y

clearly satisfies condition (i) of Theorem 1.11, and this concludes the proof of (3.5). In a similar way, it can be shown that there exists W verifying (ii) in the statement of Theorem 1.11, and such that

\frac{∂W}{∂z_{i}} (x, y, z) = W^{(i)}(x, y, z)

for a.e. x ∈ Ω, y ∈ Y_{1}, z ∈ Y_{2}. These, together with (3.1), assert the 3-scale compactness of W^{2,p} bounded sequences.

Finally we show that any such triple (u, U, W) may be attained as a 3-scale limit. Assume first that we have the extra regularity Û(x, y) := U(x, y) − A(x)y ∈ C_{c}^{∞}(Ω; C_{per}^{∞}(Y_{1})) and Ŵ (x, y, z) := W (x, y, z) − C(x, y)z ∈ C_{c}^{∞}(Ω×Y_{1}; C_{per}^{∞}(Y_{2})). Extend Ŵ (x, ·, z) to ℝ^{N} periodically with period Y_{1}.

Using (1.4) it is easy to verify that the sequence

u_{ε}(x) := u(x) + ε^{2}Û (x, \frac{x}{ε}) + ε^{4}Ŵ (x, \frac{x}{ε}, \frac{x}{ε^{2}} )

satisfies

u_{ε} ⇀^{3-s} u, Du_{ε} ⇀^{3-s} Du, and D_{xx}^{2}u_{ε} ⇀^{3-s} D_{xx}^{2}u + D_{yy}^{2}Û(x, y) + D_{zz}^{2}Ŵ (x, y, z).

Since D_{yy}^{2}Û = D_{yy}^{2}U and D_{zz}^{2}Ŵ = D_{zz}^{2}W, we have obtained (1.10).

The result for arbitrary U ∈ L^{p}(Ω; W^{2,p}(Y_{1})) with U − A(x)y ∈ L^{p}(Ω; W_{per}^{2,p}(Q; ℝ^{d})) for some A ∈ L^{p}(Ω; ℝ^{d×N}), and W ∈ L^{p}(Ω×Y_{1}; W^{2,p}(Y_{2})) with W − C(x, y)z ∈ L^{p}(Ω×Q; W_{per}^{2,p}(Q; ℝ^{d})) for some C ∈ L^{p}(Ω×Q; ℝ^{d×N}), will now follow via a standard density and diagonalization argument.

4 Proof of Theorem 1.9: The Case s = 1

We divide the proof of Theorem 1.9 into a series of lemmas, the first of which establishes the regularity of f_{hom}.

Lemma 4.1. Under the hypotheses (A1), (A2) and (H4), f_{hom} : Ω × ℝ^{d×N} → [0, +∞) is continuous.

Proof. We recall that (see (1.9))

f_{hom}(x, ξ) := inf {∫_{Q} f(x, y, ξ + Dψ(y)) dy : ψ ∈ W_{per}^{1,p}(Q; ℝ^{d})} .

Fix (x, ξ) ∈ Ω × ℝ^{d×N} and consider a sequence {(x_{n}, ξ_{n})} converging to (x, ξ). Let ε > 0 and choose ψ ∈ W_{per}^{1,p}(Q; ℝ^{d}) such that

f_{hom}(x, ξ) + ε ≥ ∫_{Q} f(x, y, ξ + Dψ(y)) dy.

We then have

lim sup_{n→∞} f_{hom}(x_{n}, ξ_{n}) ≤ lim sup_{n→∞} ∫_{Q} f(x_{n}, y, ξ_{n} + Dψ(y)) dy
= ∫_{Q} f(x, y, ξ + Dψ(y)) dy ≤ f_{hom}(x, ξ) + ε,  (4.1)

where we have used (A1), (A2) and (H4).

Conversely, let ψ_{n} ∈ W_{per}^{1,p}(Q; ℝ^{d}) be such that

f_{hom}(x_{n}, ξ_{n}) + ε ≥ ∫_{Q} f(x_{n}, y, ξ + Dψ_{n}(y)) dy.

By the coercivity condition (H4), using Poincaré-Friedrichs inequality we deduce that {ψ_{n} − ∫_{Q} ψ_{n}(z) dz} is a sequence bounded in W_{per}^{1,p}(Q; ℝ^{d}), and thus, up to a subsequence not relabelled,

ψ_{n} − ∫_{Q} ψ_{n}(z) dz ⇀ ψ in W^{1,p}

for some function ψ ∈ W_{per}^{1,p}(Q; ℝ^{d}). In view of (A1), (A2), and by Theorem 2.1, we conclude that

f_{hom}(x, ξ) ≤ ∫_{Q} f(x, y, ξ + Dψ(y)) dy ≤ lim inf_{n→∞} ∫_{Q} f(x_{n}, y, ξ + Dψ_{n}(y)) dy ≤ lim inf_{n→∞} f_{hom}(x_{n}, ξ_{n}) + ε. (4.2)

Letting ε → 0^{+} in (4.1) and in (4.2) yields

lim_{n→∞} f_{hom}(x_{n}, ξ_{n}) = f_{hom}(x, ξ).

Consider a sequence {ε_{n}} converging to 0^{+} and set

F_{{ε_{n}}}(u, A) := inf {lim inf_{n→∞} F_{ε_{n}}(u, A) : u_{n} → u in L^{p}(A; ℝ^{d})}

where

F_{ε}(u, A) := ∫_{A} f (\frac{x}{ε}, \frac{x}{ε^{2}} , Du) dx

for every u ∈ W^{1,p}(Ω; ℝ^{d}) and A ∈ A(Ω).

We define

F (u, A) := inf {∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f (x, y, Du(x) + D_{y}U(x, y) + D_{z}W (x, y, z)) dx dy dz :  (4.3)
U ∈ L^{p} (Ω; W_{per}^{1,p}(Y_{1})) , W ∈ L^{p} (Ω × Y_{1}; W_{per}^{1,p}(Y_{2})) }.

Lemma 4.2. If f satisfies hypotheses (A1), (A2), (H2), (H3) and (H4), then

F_{{ε}}(u, Ω) ≤ F (u, Ω)  (4.4)

for every u ∈ W^{1,p}(Ω; ℝ^{d}).

Proof. In view of (H4) it suffices to prove (4.4) when we have the extra regularity U ∈ C_{c}^{∞} (Ω; W_{per}^{1,p}(Y_{1})) and W ∈ C_{c}^{∞} (Ω × Y_{1}; W_{per}^{1,p}(Y_{2})). Extend W (y, ·, ξ) to ℝ^{N} as a Y_{1}-periodic function, and set u_{ε}(x) := u(x) + εU (x, \frac{x}{ε}) + ε^{2}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} ). By (1.4) we have that

u_{ε} ⇀ u ∈ W^{1,p}(Ω; ℝ^{d}),

and so

F_{\{ε\}}(u, Ω) ≤ lim inf_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , Du(x) + εD_{x}U (x, \frac{x}{ε}) + D_{y}U (x, \frac{x}{ε})

+ ε^{2}D_{x}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} ) + εD_{y}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} ) + D_{z}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )) dx.

Since by (H3) and (H4) the function f(x, y, ·) is convex and has a growth of order p, it is p-Lipschitz continuous, precisely (see [Ma2])

|f(x, y, ξ) − f(x, y, ξ′)| ≤ C (1 + |ξ|^{p−1} + |ξ′|^{p−1}) |ξ − ξ′|  (4.5)

for some constant C > 0, all x ∈ ℝ^{N}, a.e. y ∈ ℝ^{N}, and all ξ, ξ′ ∈ ℝ^{d×N}. Thus

F_{\{ε\}}(u, Ω) ≤ lim inf_{ε→0+} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , Du(x) + D_{y}U (x, \frac{x}{ε}) + D_{z}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )) dx

+ lim sup_{ε→0^{+}} ∫_{Ω} C (1 + |Du(x)|^{p−1} + |εD_{x}U (x, \frac{x}{ε})|^{p−1} + |D_{y}U (x, \frac{x}{ε})|^{p−1}

+ |ε^{2}D_{x}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p−1} + |εD_{y}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p−1} + |D_{z}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p−1})

· |εD_{x}U (x, \frac{x}{ε}) + ε^{2}D_{x}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} ) + εD_{y}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )| dx

= ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f (y, z, Du(x) + D_{y}U(x, y) + D_{z}W (x, y, z)) dx dy dz,

where we have used (1.4), (A1), (A2), and the fact that, by Hölder inequality,

lim sup_{ε→0^{+}} ∫_{Ω} (1 + |Du(x)|^{p−1} + |εD_{x}U (x, \frac{x}{ε})|^{p−1} + |D_{y}U (x, \frac{x}{ε})|^{p−1}

+ |ε^{2}D_{x}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p−1} + |εD_{y}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p−1} + |D_{z}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p−1})

· |εD_{x}U (x, \frac{x}{ε}) + ε^{2}D_{x}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} ) + εD_{y}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )| dx

≤ lim sup_{ε→0^{+}} Cε [∫_{Ω} (1 + |Du(x)|^{p} + |D_{x}U (x, \frac{x}{ε})|^{p} + |D_{y}U (x, \frac{x}{ε})|^{p}

+ |D_{x}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p} + |D_{y}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p} + |D_{z}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p}) dx]^{\frac{p}{p−1}}

· [∫_{Ω} (|D_{x}U (x, \frac{x}{ε})|^{p} + |D_{x}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p} + |D_{y}W (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p}) dx]^{\frac{1}{p}}

which converges to zero by virtue of (1.4). This concludes the proof.

Next we want to establish

Lemma 4.3. If f satisfies hypotheses (H1) [or (A1), (A2), (A3)], (H2), (H3) and (H4), then

F_{{ε}}(u, Ω) ≥ F (u, Ω)  (4.6)

for every u ∈ W^{1,p}(Ω; ℝ^{d}).

The proof of Lemma 4.3 is hinged on the two general multiscale lowersemicontinuity results below, the first of which is proven following step by step Allaire’s argument in the proof of Theorem 1.8 in [A]).

Lemma 4.4. Let 1 ≤ p ≤ +∞. If f satisfies hypotheses (A1), (A2), (A3), (H2), (H3), and (H4) if 1 ≤ p < +∞, and if a sequence {w_{e}} ⊂ L^{p}(Ω; ℝ^{d}) 3-scale converges to a function w_{0} ∈ L^{p} (Ω × Y_{1} × Y_{2}), then

lim inf_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, …, \frac{x}{ε^{2}} , w_{ε}(x)) dx = ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f(y, z, w_{0}(x, y, z)) dx dy dz.  (4.7)

Proof. As it is well known, if 1 ≤ p < +∞ and if (H4) holds then we have

|\frac{∂f}{∂ξ} (x, y, ξ)| ≤ C(1 + |ξ|^{p−1})  (4.8)

for all x ∈ ℝ^{N}, a.e. y ∈ ℝ^{N}, and all ξ ∈ ℝ^{d×N}. Indeed,

f(x, y, ξ′) ≥ f(x, y, ξ) + \frac{∂f}{∂ξ} (x, y, ξ) · (ξ′ − ξ)

and taking ξ′ := ξ ± E_{i}, with i = 1, …, d × N, and {E_{1}, …, E_{d×N}} an orthonormal basis of ℝ^{d×N}, we have by (4.5) and in view of (H4)

\frac{∂f}{∂ξ} (x, y, ξ) · (±E_{i}) ≤ f(x, y, ξ ± E_{i}) − f(x, y, ξ) ≤ C (1 + |ξ ± E_{i}|^{p−1} + |ξ|^{p−1}) ,

and (4.8) follows.

Let {θ_{j}} ⊂ C_{c} (Ω; C_{per} (Y_{1} × Y_{2})) be such that

θ_{j} → u_{0} in L^{s} (Ω × Y_{1} × Y_{2}) ,  (4.9)

where s := p′ if 1 ≤ p < +∞ and s := 1 if p = +∞ (see Remark 1.5). By the convexity of f we now have for fixed j ∈ ℕ

lim_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , w_{ε}) dx ≥ lim_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , θ_{j} (x, \frac{x}{ε}, \frac{x}{ε^{2}} )) dx

− lim sup_{ε→0^{+}} ∫_{Ω} \frac{∂f}{∂ξ} (\frac{x}{ε}, \frac{x}{ε^{2}} , θ_{j} (x, \frac{x}{ε}, \frac{x}{ε^{2}} )) · [w_{ε}(x) − θ_{j} (x, \frac{x}{ε}, \frac{x}{ε^{2}} )] dx

≥ ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f (y, z, θ_{j}(x, y, z))) dx dy dz − lim sup_{ε→0^{+}} I_{j,ε},

where we have used (A1), (A2), (A3), (H2) and (1.4) to obtain the iterated integral (note that here, and in light of Remark 1.10, the continuity of \frac{∂f}{∂ξ} is crucial), and where

I_{j,ε} := ∫_{Ω} \frac{∂f}{∂ξ} (\frac{x}{ε}, \frac{x}{ε^{2}} , θ_{j} (x, \frac{x}{ε}, \frac{x}{ε^{2}} )) · w_{ε}(x) dx − ∫_{Ω} \frac{∂f}{∂ξ} (\frac{x}{ε}, \frac{x}{ε^{2}} , θ_{j} (x, \frac{x}{ε}, \frac{x}{ε^{2}} )) · θ_{j} (x, \frac{x}{ε}, \frac{x}{ε^{2}} ) dx.

By the definition of 3-scale convergence (see Definition (1.2)) and in light of Remark 1.3, the first term in this expression converges, as ε → 0^{+}, to

∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} \frac{∂f}{∂ξ} (y, z, θ_{j}(x, y, z)) · w_{0}(x, y, z) dx dy dz,

and by (1.4) the second term tends to

∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} \frac{∂f}{∂ξ} (y, z, θ_{j}(x, y, z)) · θ_{j}(x, y, z) dx dy dz.

Given the arbitrariness of j, we conclude that

lim_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , w_{ε}) dx ≥ lim_{j→∞} ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f (y, z, θ_{j}(x, y, z)) dx dy dz

+ lim_{j→∞} ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} \frac{∂f}{∂ξ} (y, z, θ_{j}(x, y, z)) · (w_{0}(x, y, z) − θ_{j}(x, y, z)) dx

= ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f (y, z, w_{0}(x, y, z)) dx dy dz,

where to assert the equality above we have used (H4), Hölder inequality with respect to s, (4.8), (4.9), and this entails (4.7).

Lemma 4.5. If f satisfies hypotheses (H1), (H2), (H3), and if a sequence {w_{ε}} ⊂ L^{∞}(Ω; ℝ^{d}) 3-scale converges to a function w_{0} ∈ L^{∞} (Ω × Y_{1} × Y_{2}; ℝ^{d}), then

lim inf_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, …, \frac{x}{ε^{2}} , w_{ε}(x)) dx ≥ ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f(y, z, w_{0}(x, y, z)) dx dy dz.

Proof. Note that in order to be in a position to apply Lemma 4.4 we need to modify f so as to guarantee the differentiability with respect to ξ requested in (A3). Let ρ_{k}, k ∈ ℝ^{N}, denote the standard radially symmetric mollifiers. Define for k ∈ ℝ^{N}

f_{k}(x, y, ξ) := ∫_{B_{1/k}(0)} ρ_{k}(ξ′)f(x, y, ξ − ξ′) dξ′, (x, y, ξ) ∈ ℝ^{N} × ℝ^{N} × ℝ^{d×N}.

It is clear that f_{k} satisfies (H1), (H2), (H3), and (A3). Fix M > 0 such that ∥w_{ε}∥_{L^{∞}(Ω)}, ∥w_{0}∥_{L^{∞}(Ω×Y_{1}×Y_{2})} ≤ M for all ε > 0. Fix δ > 0 and choose k ∈ ℕ such that

max_{x∈Y_{1},y∈Y_{2},ξ,ξ′∈\overline{B(0,M)}} |f(x, y, ξ) − f(x, y, ξ′)| < δ whenever |ξ − ξ′| ≤ \frac{1}{k} .

Since f(·, ·, ξ) is periodic, we then have

sup_{x,y∈ℝ^{N} ,ξ,ξ′∈\overline{B(0,M)}} |f(x, y, ξ) − f(x, y, ξ′)| < δ whenever |ξ − ξ′| ≤ \frac{1}{k} ,

and thus

sup_{x,y∈ℝ^{N} ,ξ∈\overline{B(0,M)}} |f_{k}(x, y, ξ) − f(x, y, ξ′)| ≤ δ.

By Lemma 4.4 we have

lim_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , w_{ε}) dx ≥ lim_{ε→0^{+}} ∫_{Ω} f_{k} (\frac{x}{ε}, \frac{x}{ε^{2}} , w_{ε}) dx − δ
≥ ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f_{k}(y, z, w_{0}(x, y, z)) dx dy dz − δ
≥ ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f(y, z, w_{0}(x, y, z)) dx dy dz − 2δ.

Now it suffices to let δ → 0^{+}.

Proof of Lemma 4.3. Step 1. We assume first that f satisfies (A1), (A2), (A3), (H2), (H3) and (H4). Let u_{ε} ⇀ u in W^{1,p}(Ω; ℝ^{d}). Extracting a subsequence if necessary, without loss of generality and by Theorem 1.7 we may assume that

u_{ε} ⇀^{3-s} u, Du_{ε} ⇀^{3-s} Du + D_{y}U(x, y) + D_{z}W (x, y, z)

for some U ∈ L^{p} (Ω; W_{per}^{1,p}(Y_{1})) and W ∈ L^{p} (Ω × Y_{1}; W_{per}^{1,p}(Y_{2})). We are in a position to apply Lemma 4.4, where (4.7) becomes

lim inf_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , Du_{ε}) dx ≥ ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f (y, z, Du(x) + D_{y}U(x, y) + D_{z}W (x, y, z)) dx dy dz
≥ F (u, Ω).

Step 2. Now assume that f satisfies (H1), (H2), (H3) and (H4) (but not necessarily (A3)), and u_{ε} ⇀ u in W^{1,p}(Ω; ℝ^{d}). By Proposition 2.5 we may find a subsequence (not relabelled) and a new sequence {v_{ε}} (if p = 1 or p = +∞ we set v_{ε} := u_{ε}), which still converges weakly in W^{1,p}(Ω; ℝ^{d}) to u and {|Dv_{ε}|^{p}} is equi-integrable. If 1 < p < +∞ we have by (H4) and

lim inf_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , Du_{ε}) dx ≥ lim inf_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , Dv_{ε}) dx  (4.10)
− C lim sup_{ε→0^{+}} ∫_{{u_{ε}≠v_{ε}}}(1 + |Dv_{ε}|^{p}) dx  (4.11)
= lim inf_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , Dv_{ε}) dx,  (4.12)

where we have used the equi-integrability property and (2.3).

Extracting a subsequence if necessary, without loss of generality and by Theorem 1.7 we may assume that

v_{ε} ⇀^{3-s} u, Dv_{ε} ⇀^{3-s} T := Du + D_{y}U(x, y) + D_{z}W (x, y, z),

and

τ_{M} (Dv_{ε}) ⇀^{3-s} T_{M}

for all M ∈ ℕ, and for some U ∈ L^{p} (Ω; W_{per}^{1,p}(Y_{1})), W ∈ L^{p} (Ω × Y_{1}; W_{per}^{1,p}(Y_{2})), and T_{M} ∈ L^{p}(Ω × Y_{1} × Y_{2}; ℝ^{d×N}). Here τ_{M}(·) stands for the usual truncation function

τ_{M}(ξ) := \{ \begin{matrix}ξ & if |ξ| ≤ M, \\ M \frac{ξ}{|ξ|} & otherwise. \end{matrix}

Note that

sup_{M∈ℕ} ∥T_{M}∥_{L^{p}(Ω×Y_{1}×Y_{2})} ≤ sup_{M∈ℕ} lim inf_{ε>0} ∥τ_{M} (Dv_{ε}) ∥_{L^{p}(Ω)} ≤ sup_{ε>0} ∥Dv_{ε}∥_{L^{p}(Ω)} < +∞,

and thus every subsequence of {T_{M}} admits a further subsequence converging weakly in L^{p}(Ω × Y_{1} × Y_{2}; ℝ^{d×N}). We claim that the weak limit is T, thus asserting that the full sequence {T_{M}} converges to T weakly in L^{p}(Ω × Y_{1} × Y_{2}). Indeed, fix φ ∈ C_{c}^{∞}(Ω; C_{per}^{∞}(Y_{1} × Y_{2})). We have

lim_{M→+∞} ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} φ(x, y, z) T_{M}(x, y, z) dx = lim_{M→+∞} lim_{ε→0^{+}} ∫_{Ω} [∫_{Ω} φ (x, \frac{x}{ε}, \frac{x}{ε^{2}} ) Dv_{ε}(x) dx

− ∫_{\{|Dv_{e}|>M\}} φ (x, \frac{x}{ε}, \frac{x}{ε^{2}} ) (Dv_{e} − τ_{M}(Dv_{e})) dx]

= ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} φ(x, y, ξ)T (x, y, ξ) dx dy dz,

where we have used (1.2), and where due to (1.4) and to the equi-integrability of {|Dv_{ε}|^{p}}, by Hoölder inequality we deduce

lim sup_{M→+∞} lim sup_{ε→0^{+}} ∫_{\{|Dv_{ε}|>M\}} |φ (x, \frac{x}{ε}, \frac{x}{ε^{2}} )| |Dv_{ε} − τ_{M}(Dv_{ε})| dx

≤ C [ lim_{ε→0^{+}} (∫_{Ω} |φ (x, \frac{x}{ε}, \frac{x}{ε^{2}} )|^{p′} dx)^{1/p′} lim sup_{M→+∞} sup_{ε>0^{+}} (∫_{\{|Dv_{ε}|>M\}} |Dv_{ε}|^{p} dx)^{1/p}]

= (∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} |φ(x, y, z)| dx dy dz) (lim sup_{M→+∞} sup_{ε>0^{+}} (∫_{{|Dv_{ε}|>M}} |Dv_{ε}|^{p} dx)^{1/p}) = 0.

By Lemma 4.5 and for every fixed M ∈ ℕ we have

lim inf_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , Dv_{ε}) dx ≥ lim inf_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , τ_{M}(Dv_{ε})) dx

− lim sup_{ε→0^{+}} ∫_{\{|Dv_{ε}|>M\}} f (\frac{x}{ε}, \frac{x}{ε^{2}} , τ_{M}(Dv_{ε})) dx

≥ ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f(y, z, T_{M}(x, y, z)) dx dy dz − sup_{ε>0} C ∫_{{|Dv_{ε}|>M}}(1 + |Dv_{ε}|^{p}) dx,

where we have used (H4). Letting M → +∞, since T_{M} ⇀ T in L^{p} by Theorem 2.1 and in view of the equi-integrability of {|Dv_{ε}|^{p}} we obtain

lim inf_{ε→0^{+}} ∫_{Ω} f (\frac{x}{ε}, \frac{x}{ε^{2}} , Dv_{ε}) dx ≥ ∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f(y, z, T (x, y, z)) dx dy dz,

which, together with (4.10), concludes the proof.

Lemma 4.6. If f satisfies hypotheses (H1) [or (A1), (A2)], (H2), (H3) and (H4), then

F_{\{ε\}}(u, Ω) ≤ ∫_{Ω} \overline{f_{hom}}(Du(x)) dx  (4.13)

for every u ∈ W^{1,p}(Ω; ℝ^{d}).

Proof. Fix u ∈ W^{1,p}(Ω; ℝ^{d}). In view of (H4) it can be shown easily that F_{{ε}}(u, ·) is the trace on A(Ω) of a Radon measure absolutely continuous with respect to the N-dimensional Lebesgue measure L^{N} (for proofs of similar results see [BFLM], [BrFF]). Therefore, proving (4.13) is equivalent to showing that

lim_{δ→0^{+}} \frac{F_{\{ε\}}(u, Q(x_{0}, δ))}{δ^{N}} ≤ \overline{f_{hom}}(Du(x_{0}))

for a.e. x_{0} ∈ Ω. Choose x_{0} to be a p-Lebesgue point for Du, i.e.

lim_{δ→0^{+}} \frac{1}{δ^{N}} ∫_{Q(x_{0},δ)} |Du(x) − Du(x_{0})|^{p} dx = 0,  (4.14)

fix α > 0, and using the definition of \overline{f_{hom}}(Du(x_{0})) choose φ ∈ C_{per}^{∞}(Q; ℝ^{d}) such that

\overline{f_{hom}}(Du(x_{0})) + α ≥ ∫_{Q} f_{hom}(x, Du(x_{0}) + Dφ(x)) dx.  (4.15)

In order to apply Theorem 2.3, with (X, M) := (Ω, L), Z := W_{per}^{1,p}(Q; ℝ^{d}), μ is the Lebesgue measure, and L is the σ-algebra of Lebesgue measurable sets in ℝ^{N}, we introduce the multi-valued map

H : Ω → {C ⊂ W_{per}^{1,p}(Q) : C ≠ Ø, C is closed}
x ↦H(x) := {ψ ∈ W_{per}^{1,p}(Q; ℝ^{d}) : ∫_{Q} ψ(y) dy = 0,

f_{hom}(x, Du(x_{0}) + Dφ(x)) + α ≥ ∫_{Q} f(x, y, Du(x_{0}) + Dφ(x) + Dψ(y)) dy} .

By definition of f_{hom}, the set H(x) is nonnempty and by (H4) it is closed. Also, the set

{(x, ψ) ∈ Ω × W_{per}^{1,p}(Q; ℝ^{d}) : ψ ∈ H(x)}

is closed (hence Borel) because it coincides with H^{−1}([0, +∞) × {0}) where

H(x, ψ) := (f_{hom}(x, Du(x_{0}) + Dφ(x)) + α − ∫_{Q} f(x, y, Du(x_{0}) + Dφ(x) + Dψ(y)) dy, ∫_{Q} ψ(y) dy) ,

and by (H4), and in view of the continuity of Dφ and of f_{hom} (see Lemma 4.1), the function H is continuous. By Theorem 2.3 we may find a measurable selection

x ↦ ψ(x, ·), ψ(x, ·) ∈ W_{per}^{1,p}(Q; ℝ^{d}), ∫_{Q} ψ(x, y) dy = 0,

and

f_{hom}(x, Du(x_{0}) + Dφ(x)) + α ≥ ∫_{Q} f(x, y, Du(x_{0}) + Dφ(x) + D_{y}ψ(x, y)) dy.  (4.16)

By (H4) and (4.16) we now have

+ ∞ > ∫_{Q} [f_{hom}(x, Du(x_{0}) + Dφ(x)) + α] dx

≥ C ∫_{Q} \{∫_{Q} [|D_{y}ψ(x, y)|^{p} − |Du(x_{0})|^{p} − |Dφ(x)|^{p}] dy\} dx ≥ C′ ∫_{Q} ∥ψ(x, ·)∥_{W_{per}^{1,p}(Q)}^{p} dx − \frac{1}{C′} ,

where we have used Poincaré-Friedrichs inequality. We conclude that ψ ∈ L^{p} (Q; W_{per}^{1,p}(Q; ℝ^{d})). Now let ψ_{k} ∈ C_{c}^{∞} (Q; W_{per}^{1,p}(Q; ℝ^{d})) be such that

∥ψ_{k} − ψ∥_{L^{p}(Q;W_{per}^{1,p}(Q;ℝ^{d}))} → 0,  (4.17)

and define

u_{k,ε} := u(x) + εφ (\frac{x}{ε}) + ε^{2}ψ_{k} (\frac{x}{ε}, \frac{x}{ε^{2}} ) .

For fixed δ > 0 it is clear that u_{k,ε} → u in L^{p}(Q(x_{0}, δ)), and so

lim_{δ→0^{+}} \frac{F_{\{ε\}}(u, Q(x_{0}, δ))}{δ^{N}}

≤ lim inf_{δ→0^{+}} lim_{ε→0^{+}} \frac{1}{δ^{N}} ∫_{Q(x_{0},δ)} f (\frac{x}{ε}, \frac{x}{ε^{2}} , Du(x) + Dφ (\frac{x}{ε}) + εD_{x}ψ_{k} (\frac{x}{ε}, \frac{x}{ε^{2}} ) + D_{y}ψ_{k} (\frac{x}{ε}, \frac{x}{ε^{2}} )) dx

≤ lim_{δ→0^{+}} lim_{ε→0^{+}} \frac{1}{δ^{N}} ∫_{Q(x_{0},δ)} f (\frac{x}{ε}, \frac{x}{ε^{2}} , Du(x_{0}) + Dφ (\frac{x}{ε}) + D_{y}ψ_{k} (\frac{x}{ε}, \frac{x}{ε^{2}} )) dx

+ C lim sup_{δ→0^{+}} lim sup_{ε→0^{+}} \frac{1}{δ^{N}} [∫_{Q(x_{0},δ)} (1 + |Du(x)|^{p} + |Dφ (\frac{x}{ε})|^{p}

+ |εD_{x}ψ_{k} (\frac{x}{ε}, \frac{x}{ε^{2}} )|^{p} + |D_{y}ψ_{k} (\frac{x}{ε}, \frac{x}{ε^{2}} )|^{p}) dx]^{\frac{p−1}{p}}

· [∫_{Q(x_{0},δ)} (|Du(x) − Du(x_{0})|^{p} + |εD_{x}ψ_{k} (\frac{x}{ε}, \frac{x}{ε^{2}} )|^{p}) dx]^{\frac{1}{p}}

= ∫_{Y_{1}} ∫_{Y 2} f(x, y, Du(x_{0}) + Dφ(x) + D_{y}ψ_{k}(x, y)) dx dy

where we have used (4.5), (1.4), (4.14) and Hölder inequality. We deduce that

lim_{δ→0^{+}} \frac{F_{\{ε\}}(u, Q(x_{0}, δ))}{δ^{N}} ≤ ∫_{Y_{1}} ∫_{Y 2} f(x, y, Du(x_{0}) + Dφ(x) + D_{y}ψ_{k}(x, y)) dx dy,

and in view of (H4) and by (4.15), (4.16) and (4.17), letting k → +∞ we obtain

lim_{δ→0^{+}} \frac{F_{\{ε\}}(u, Q(x_{0}, δ))}{δ^{N}} ≤ ∫_{Y_{1}} ∫_{Y 2} f(x, y, Du(x_{0}) + Dφ(x) + D_{y}ψ(x, y)) dx dy
≤ \overline{f_{hom}}(Du(x_{0})) + 2α.

Letting α → 0^{+} we conclude (4.13).

Proof of Theorem 1.9. Fix u ∈ W^{1,p}(Ω; ℝ^{d}) and let {ε_{n}} be a sequence converging to 0^{+}. By Theorem 2.2 we may extract a subsequence {ε_{j}} such that the Γ(L^{p}(Ω)) − lim_{j→∞} F_{ε_{j}}(u, A) exists and is given by F_{{ε_{j}}}. As before, for simplicity of notation we write simple ε in place of ε_{j}. We claim that

F_{\{ε_{j}\}}(u, A) = ∫_{A} \overline{f_{hom}}(Du(x)) dx.

If the claim is asserted, then due to the arbitrariness of the initial sequence {ε_{n}} we conclude the proof of Theorem 1.9. Now Lemma 4.6 establishes that

F_{\{ε_{j}\}}(u, A) ≤ ∫_{A} \overline{f_{hom}}(Du(x)) dx,

and in view of Lemma 4.3, it suffices to prove that

F (u, A) ≥ ∫_{A} \overline{f_{hom}}(Du(x)) dx.

Let U ∈ L^{p} (Ω; W_{per}^{1,p}(Y_{1})) and W ∈ L^{p} (Ω × Y_{1}; W_{per}^{1,p}(Y_{2})). By the very definitions of f_{hom} and \overline{f_{hom}} (see (1.9) and (1.8)) we have by Fubini’s Theorem

∫_{A} ∫_{Y_{1}} ∫_{Y_{2}} f(y, z, Du(x) + D_{y}U(x, y) + D_{z}W (x, y, z)) dx dy dz
≥ ∫_{A} ∫_{Y_{1}} [∫_{Y_{2}} f(y, z, Du(x) + D_{y}U(x, y) + D_{z}W (x, y, z)dz] dy dx
≥ ∫_{A} ∫_{Y_{1}} f_{hom}(y, Du(x) + D_{y}U(x, y)) dx dy ≥ ∫_{A} \overline{f_{hom}}(Du(x))dx,

and so

F (u, A) ≥ ∫_{Ω} \overline{f_{hom}}(Du(x)) dx.

5 Proof of Theorem 1.9: The Case s = 2

The proof of Theorem 1.9 for second order derivatives follows closely that of the case where s = 1 treated in the previous section, with the obvious adaptations. In particular, in Lemma 4.3 the compactness with respect to 3-scale convergence should now invoke Theorem 1.11 in place of Theorem 1.7, and the function F introduced in (4.3) should be replaced by

F (u, A) := inf {∫_{Ω} ∫_{Y_{1}} ∫_{Y_{2}} f(x, y, D_{xx}^{2}u(x) + D_{yy}^{2}U(x, y) + D_{zz}^{2}W (x, y, z)) dx dy dz :
U ∈ L^{p}(Ω; W_{per}^{2,p}(Y_{1})), W ∈ L^{p}(Ω × Y_{1}; W_{per}^{2,p}(Y_{2})),
U − A(x)y ∈ L^{p}(Ω; W_{per}^{2,p}(Q; ℝ^{d})) for some A ∈ L^{p}(Ω; ℝ^{d×N}),
W − C(x, y)z ∈ L^{p}(Ω×Q; W_{per}^{2,p}(Q; ℝ^{d})) for some C ∈ L^{p}(Ω×Q; ℝ^{d×N})}.

We leave the details to the reader.

6 Appendix

Here we present the proof of Proposition 2.2, which, in turn, uses the following result (see Theorem 2.2. in [BhFF] and Theorem 2.5 in [BrFF]).

Proposition 6.1. Let F_{ε} : W^{1,p}(Ω; ℝ^{d})×A(Ω) → [0, +∞) be a sequence of functionals verifying the assumptions (i)–(iii) of Proposition 2.2. Then for any u ∈ W^{1,p}(Ω, ℝ^{d})

Γ^{p}(u, A) := Γ(L^{p}) − lim inf_{ε→0} F_{ε}(u, A) = inf {lim inf_{ε→0} F_{ε}(u_{ε}, A) : u_{ε}→u in L^{p}(A)}

coincides with

Γ_{0}^{p}(u, A) := inf {lim inf_{ε→0} F_{ε}(u_{ε}, A) : u_{ε}→u in L^{p}(A), u_{ε} = u on a neighborhood of ∂A} .

Proof of Proposition 6.1. Clearly Γ^{p}(u, A) ≤ Γ_{0}^{p}(u, A), so it remains to prove the opposite inequality. Define for every u ∈ W^{1,p}(Ω; ℝ^{d}) the functional G(u, A) := ∫_{A} (1 + |Du|^{p}) dx. For every γ > 0 there exists a sequence {u_{ε}} ⊂ W^{1,p}(A, ℝ^{m}) such that u_{ε}→u in L^{p}(A, ℝ^{d}) and lim inf_{ε} F_{ε}(u_{ε}, A) ≤ Γ^{p}(u, A) + γ. By the coercivity assumption (2.1) there exists a subsequence {u_{ε_{k}}} of {u_{ε}} such that lim_{k→∞} F_{ε_{k}}(u_{ε_{k}}, A) = lim inf_{ε→0^{+}} F_{ε}(u_{ε}, A) and the sequence of measures ν_{k} := G(u, ·) + G(u_{ε_{k}}, ·) ⇀^{∗} ν, where ν is a finite Radon measure.

For every t > 0 let A_{t} := {x ∈ A : dist(x, ∂A) > t}, fix η > 0 and for any 0 < δ < η define L_{δ} := A_{η−2δ}\\overline{A_{η+δ}}. Next, Consider a smooth cut off function φ_{δ} ∈ C_{c}^{∞}(A_{η−δ}; [0, 1]) such that φ_{δ} = 1 on A_{η}, ∥Dφ_{δ}∥_{∞} ≤ \frac{C}{δ} and set w_{ε} := φ_{δ}u_{ε} + (1 − φ_{δ})u. Then w_{ε}→u in L^{p}(A, ℝ^{d}) as ε → 0^{+} and w_{ε} = u on a neighborhood of ∂A. Thus,

F_{ε_{k}}(w_{ε_{k}}, A) ≤ F_{ε_{k}}(w_{ε_{k}}, A_{η}) + F_{ε_{k}}(w_{ε_{k}}, A\A_{η−δ}) + F_{ε_{k}}(w_{ε_{k}}, A_{η−2δ}\\overline{A_{η+δ}})

≤ F_{ε_{k}}(u_{ε_{k}}, A_{η}) + F_{ε_{k}}(u, A\\overline{A_{η−δ}}) + C [G(u_{ε_{k}}, L_{δ}) + G(u, L_{δ}) + \frac{1}{δ^{p}} ∫_{L_{δ}} |u_{ε_{k}} − u|^{p} dx] .

Letting ε_{k}→0^{+} we get

lim sup_{ε_{k}→0^{+}} F_{ε_{k}}(w_{ε_{k}}, A) ≤ Γ^{p}(u, A) + Cν(A\\overline{A_{η−δ}}) + Cν(\overline{L_{δ}}) + γ,

and so

Γ_{0}^{p}(u, A) ≤ Γ^{p}(u, A) + γ + Cν(A\\overline{A_{η−δ}}) + Cν(\overline{L_{δ}}).

Letting δ go to 0^{+} we have

Γ_{0}^{p}(u, A) ≤ Γ^{p}(u, A) + γ + Cν(A\A_{η}) + Cν(∂A_{η}).

Choosing a sequence {η_{j}} such that η_{j}→0^{+} and ν(∂A_{η_{j}}) = 0, letting first j→ + ∞ and finally γ→0^{+}, now yields

Γ_{0}^{p}(u, A) ≤ Γ^{p}(u, A).

Proof of Proposition 2.2. We consider a countable collection C of open subsets of Ω such that for all δ > 0 and A ∈ A(Ω) there exists a finite union C_{A} of disjoint elements of C such that

\{ \begin{matrix}\overline{C_{A}} ⊂ A \\ L^{N}(A) ≤ L^{N}(C_{A}) + δ. \end{matrix}

Denote by R the countable collection of all finite unions of elements of C, i.e. R := {∪_{i=1}^{k}C_{i} : k ∈ ℕ, C_{i} ∈ C}. Since L^{p} is a separable metric space, by using a diagonalization argument we can assert that there exists a subsequence {ε_{R}} ⊂ {ε}, ε_{R} → 0, such that for every C ∈ R and for every v ∈ W^{1,p}(C; ℝ^{d}) there exists {u_{ε_{R}}^{C}} in W^{1,p}(C; ℝ^{d}) such that u_{ε_{R}}^{C}→v in L^{p}(C; ℝ^{d}) and

Γ(L^{p}(C)) − lim_{ε_{R}→0^{+}} F_{ε_{R}}(v, C) = lim_{ε_{R}→0^{+}} F_{ε_{R}}(u_{ε_{R}}^{C}, C).  (6.1)

In other words, we get the existence of the Γ−limit of a fixed subsequence and for every C ∈ R. We seek to extend (6.1) from the elements of R to any open subset A of Ω and every u ∈ W^{1,p}(A; ℝ^{d}). Let A be an open subset of Ω and u ∈ W^{1,p}(A; ℝ^{d}). Fix δ > 0 and choose a subset C^{δ} of A in R such that

\{ \begin{matrix}\overline{C^{δ}} ⊂ A, \\ ∫_{A\C^{δ}} (1 + |Du|^{p}) dx ≤ \frac{δ}{C} , \end{matrix}

where C is the constant of the growth condition (2.1). Consider a sequence {v_{ε^{R}}^{C^{δ}}} in W^{1,p}(C^{δ}; ℝ^{d}) such that v_{ε^{R}}^{C^{δ}}→u in L^{p}(C^{δ}; ℝ^{d}) satisfying

lim_{ε^{R}→0^{+}} F_{ε^{R}}(v_{ε^{R}}^{C^{δ}}, C^{δ}) = F_{ε^{R}}^{−}(u, C^{δ}) = Γ(L^{p}(C)) − lim_{ε^{R}→0^{+}} F_{ε^{R}}(u, C^{δ}).

By Proposition 6.1 we may assume that v_{ε_{R}}^{C^{δ}} = u nearby ∂C^{δ}, and extend v_{ε^{R}}^{C^{δ}} as u outside C^{δ}, so that the extension (not relabeled) v_{ε^{R}}^{C^{δ}} belongs to W^{1,p}(A; ℝ^{d}). We have

lim sup_{δ→0^{+}} lim sup_{ε^{R}→0^{+}} F_{ε^{R}}(v_{ε^{R}}^{C^{δ}}, A) ≤ lim sup_{δ→0^{+}} lim sup_{ε^{R}→0^{+}} {F_{ε^{R}}(v_{ε^{R}}^{C^{δ}}, C^{δ}) + C ∫_{A\C^{δ}} (1 + |Du|^{p}) dx}
= lim sup_{δ→0^{+}} F_{{ε^{R}}}(u, C^{δ}) ≤ F_{{ε^{R}}}(u, A) ≤ lim inf_{δ→0^{+}} lim inf_{ε^{R}→0^{+}} F_{ε^{R}}(v_{ε^{R}}^{C^{δ}}, A). (6.2)

A diagonalization argument (cf. Lemma 7.1 [BrFF]) and (6.2) concludes the proof.

Acknowledgments

The research of I. Fonseca was partially funded by the National Science Foundation under Grant DMS– 0103798, the research of E. Zappale by Università di Salerno through PGR Omogeneizzazione di materiali perforati and by GNAMPA. The authors thank the Center for Nonlinear Analysis (NSF Grant No. DMS– 9803791), Carnegie Mellon University, Pittsburgh, PA, USA, for its support during the preparation of this paper. E. Zappale is indebted to the Department of Mathematical Sciences at Carnegie Mellon University for its hospitality during her residence at CMU in 2000/01.

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