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A-QUASICONVEXITY: RELAXATION AND HOMOGENIZATION

Andrea Braides^{1}, Irene Fonseca^{2} and Giovanni Leoni ^{3}

Abstract. Integral representation of relaxed energies and of Γ-limits of functionals

(u, v) ↦ ∫_{Ω} f(x, u(x), v(x)) dx

are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell’s equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W^{1,p} are recovered.

AMS Subject Classification. 35D99, 35E99, 49J45.

Received March 16, 2000. Revised September 26, 2000.

1. Introduction

In a recent paper Fonseca and Müller [22] have proved that A-quasiconvexity is a necessary and sufficient condition for (sequential) lower semicontinuity of a functional

(u, v) ↦ ∫_{Ω} f(x, u(x), v(x)) dx,

whenever f : Ω × ℝ^{m} × ℝ^{d} → [0, ∞) is a Carathéodory integrand satisfying

0 ≤ f(x, u, v) ≤ a(x, u) (1 + |v|^{q}) ,

for a.e. x ∈ Ω and all (u, v) ∈ ℝ^{m} × ℝ^{d}, where 1 ≤ q < ∞, a ∈ L_{loc}^{∞}(Ω × ℝ; [0, ∞)), Ω ⊂ ℝ^{N} is open, bounded, u_{n} → u in measure, v_{n} ⇀ v in L^{q}(Ω; ℝ^{d}) and Av_{n} → 0 in W^{−1,q}(Ω; ℝ^{l}) (see also [14]). Here, and in what follows, following [32]

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

\begin{matrix}c \\ ◯ \end{matrix} EDP Sciences, SMAI 2000

Keywords and phrases: A-quasiconvexity, equi-integrability, Young measure, relaxation, Γ-convergence, homogenization.

1 SISSA, Trieste, Italy.

2 Department of MathematicalSciences, Carnegie-Mellon University, Pittsburgh, PA, U.S.A.; e-mail: Fonseca@anderw.cmu.edu

3 Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy.

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

rank 𝔸w = r for all w ∈ S^{N−1},  (1.1)

where

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

A function f : ℝ^{d} → ℝ is said to be A-quasiconvex if

f(v) ≤ ∫_{Q} f(v + w(y)) dy

for all v ∈ ℝ^{d} and all w ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{d}) such that Aw = 0 and ∫_{Q} w(y) dy = 0. Here Q denotes the unit cube in ℝ^{N}, and the space C_{1-per}^{∞}(ℝ^{N}; ℝ^{d}) is introduced in Section 2.

The relevance of this general framework, as emphasized by Tartar (see [32, 34–39]), 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. Some important examples included in this general setting are given by:

(a) [Unconstrained Fields]

Av ≡ 0.

Here, due to Jensen’s inequality A-quasiconvexity reduces to convexity.

(b) [Divergence Free Fields] Av = 0 if and only if div v = 0,

where v : Ω ⊂ ℝ^{N} → ℝ^{N} (see [33]).

(c) [Magnetostatics Equations]

A (\begin{matrix}m \\ h \end{matrix}) := (\begin{matrix}div(m + h) \\ curl h \end{matrix}) = 0,

where m : ℝ^{3} → ℝ^{3} is the magnetization and h : ℝ^{3} → ℝ^{3} is the induced magnetic field (see [17,38]); often these are also called Maxwell’s Equations in the micromagnetics literature.

(d) [Gradients] Av = 0 if and only if curl v = 0.

Note that w ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{d}) is such that curl w = 0 and ∫_{Q} w(y) dy = 0 if and only if there exists φ ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{n}) such that ∇φ = v, where d = n × N. Thus in this case we recover the well-known notion of quasiconvexity introduced by Morrey [30].

(e) [Higher Order Gradients] Replacing the target space ℝ^{d} by an appropriate finite dimensional vector space E_{s}^{n}, it is possible to find a first order linear partial differential operator A such that v ∈ L^{p}(Ω; E_{s}^{n}) and Av = 0 if and only if there exists φ ∈ W^{s,q}(Ω; ℝ^{n}) such that v = ∇^{s}φ (see Th. 1.3).

This paper is divided into two parts. In the first part we give an integral representation formula for the relaxed energy in the context of A–quasiconvexity. Precisely, let 1 ≤ p < ∞ and 1 < q < ∞, and consider the functional

F : L^{p}(Ω; ℝ^{m}) × L^{q}(Ω; ℝ^{d}) × O(Ω) → [0, ∞)

defined by

F ((u, v); D) := ∫_{D} f(x, u(x), v(x)) dx,

where O(Ω) is the collection of all open subsets of Ω, and the density f satisfies the following hypothesis:

(H) f : Ω × ℝ^{m} × ℝ^{d} → [0, ∞) is Carathéodory function satisfying

0 ≤ f(x, u, v) ≤ C (1 + |u|^{p} + |v|^{q})

for a.e. x ∈ Ω and all (u, v) ∈ ℝ^{m} × ℝ^{d}, and where C > 0.

For D ∈ O(Ω) and (u, v) ∈ L^{p}(Ω; ℝ^{m}) × (L^{q}(Ω; ℝ^{d}) ∩ ker A) define

F((u, v); D) := inf{lim inf_{n→∞} F ((u_{n}, v_{n}); D) : (u_{n}, v_{n}) ∈ L^{p}(D; ℝ^{m}) × L^{q}(D; ℝ^{d}),
u_{n} → u in L^{p}(D; ℝ^{m}), v_{n} ⇀ v in L^{q}(D; ℝ^{d}), Av_{n} → 0 in W^{−1,q}(D; ℝ^{l}) }·

(1.2)

It turns out that the condition Av_{n} → 0 imposed in (1.2) may be replaced by requiring that v_{n} do satisfy the homogeneous PDE Av = 0. Precisely, and in view of Lemma 3.1 and Corollary 3.2 below, it can be shown that

F((u, v); D) = inf {lim inf_{n→∞} F ((u, v_{n}); D) : v_{n} ∈ L^{q}(D; ℝ^{d}), v_{n} ⇀ v in L^{q}(D; ℝ^{d}), Av_{n} = 0} ,

and thus

F((u, v); D) = inf{lim inf_{n→∞} F ((u_{n}, v_{n}); D) : (u_{n}, v_{n}) ∈ L^{p}(D; ℝ^{m}) × L^{q}(D; ℝ^{d}),
u_{n} → u in L^{p}(D; ℝ^{m}), v_{n} ⇀ v in L^{q}(D; ℝ^{d}), Av_{n} = 0} =: F_{0}((u, v); D).

(1.3)

The first main result of the paper is given by the following theorem:

Theorem 1.1. Under condition (H) and the constant-rank hypothesis (1.1), for all D ∈ O(Ω), u ∈ L^{p}(Ω; ℝ^{m}), and v ∈ L^{q}(Ω; ℝ^{d}) ∩ ker A, we have

F((u, v); D) = ∫_{D} Q_{A}f(x, u(x), v(x)) dx

where, for each fixed (x, u) ∈ Ω×ℝ^{m}, the function Q_{A}f(x, u, ·) is the A-quasiconvexification of f(x, u, ·), namely

Q_{A}f(x, u, v) := inf { ∫_{Q} f(x, u, v + w(y)) dy : w ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{d}) ∩ ker A, ∫_{Q} w(y) dy = 0}

for all v ∈ ℝ^{d}.

Remarks 1.2. (i) Note that in the degenerate case where A = 0, A-quasiconvex functions are convex and Theorem 1.1 together with condition (1.4) yield a convex relaxation result with respect to L^{p} × L^{q}(weak) convergence. See the monograph of Buttazzo [12] for related results in this context.

(ii) If the function f also satisfies a growth condition of order q from below in the variable v, that is

f(x, u, v) ≥ \frac{1}{C} |v|^{q} − C  (1.4)

for a.e. x ∈ Ω and all (u, v) ∈ ℝ^{m} ×ℝ^{d}, then a simple diagonalization argument shows that (u, v) ↦ F((u, v); D) is L^{p} × (L^{q}-weak) lower semicontinuous, i.e.,

∫_{D} Q_{A}f(x, u(x), v(x)) dx ≤ lim inf_{n→∞} ∫_{D} Q_{A}f(x, u_{n}(x), v_{n}(x)) dx  (1.5)

whenever u_{n} ∈ L^{p}(Ω; ℝ^{m}), v_{n} ∈ L^{q}(Ω; ℝ^{d}) ∩ ker A, u_{n} → u in L^{p}(D; ℝ^{m}), v_{n} ⇀ v in L^{q}(D; ℝ^{d}). In particular Q_{A}f is A-quasiconvex if f is continuous and

\frac{1}{C} |v|^{q} − C ≤ f(v) ≤ C(1 + |v|^{q})

for some C > 0, and all v ∈ ℝ^{d} (see the proof of Cor. 5.7).

The lower semicontinuity result (1.5) is not covered by Theorem 3.7 in [22], where it is assumed that the integrand be A-quasiconvex and continuous in the v variable. However, as remarked in [22], in the realm of general A-quasiconvexity the function Q_{A}f(x, u, ·) may not be continuous, even if f(x, u, ·) is. Indeed in the degenerate case ker A = {0} all functions are A-quasiconvex. Also, when N = 1, d = 2, and v = (v_{1}, v_{2}), consider

Av := (0 1) (\begin{matrix}v′_{1} \\ v′_{2} \end{matrix}) .

Then for w ∈ ℝ

𝔸w = (0 w)

and thus when |w| = 1 the matrix 𝔸w has constant rank 1. For any given function f(v) the A-quasiconvex envelope of f is obtained by convexification in the first component, so that by considering e.g. (cf. [22, 28])

f_{1}(v) := e^{−|v_{1}| v_{2}^{2}}, f_{2}(v) := (1 + |v_{1}|)^{|v_{2}|},

one gets

Q_{A}f_{1}(v) = \begin{cases}0 & if v_{2} ≠ 0 \\ 1 & if v_{2} = 0 \end{cases} , Q_{A}f_{1}(v) = \begin{cases}(1 + |v_{1}|)^{|v_{2}|} & if |v_{2}| ≥ 1 \\ 1 & if |v_{2}| < 1. \end{cases}

(iii) The continuity of f with respect to v is essential to ensure the representation of F provided in Theorem 1.1, in contrast with the case where Av = 0 if and only if curl v = 0. In fact, if f : ℝ^{n×N} → [0, ∞) is a Borel function satisfying the growth condition

0 ≤ f(v) ≤ C(1 + |v|^{q})

for C > 0, 1 ≤ q < ∞, v ∈ ℝ^{n×N}, then it can be shown easily that

F(w; D) = ∫_{D} Qf(∇w(x)) dx  (1.6)

for all D ∈ O(Ω), w ∈ W^{1,q}(Ω; ℝ^{n}), where Qf is the quasiconvex envelope of f. Indeed, Qf is a (continuous) quasiconvex function satisfying (H) (see [8, 18] Th. 4.3); therefore by Theorem 1.1

w ↦ ∫_{D} Qf(∇w(x)) dx

is W^{1,q}-sequentially weakly lower semicontinuous, and so

∫_{D} Qf(∇w(x)) dx ≤ F(w; D).

Conversely, under hypothesis (H) it is known that F(v; ·) admits an integral representation (see Th. 9.1 in [10], Th. 20.1 in [15])

F(w; D) = ∫_{D} φ(∇w(x)) dx,

where φ is a quasiconvex function, and φ(v) ≤ f(v) for all v ∈ ℝ^{n×N}. Hence φ ≤ Qf and we conclude that (1.6) holds.

For general constant-rank operators A, and if f is not continuous with respect to v, it may happen that F_{0}((u, v); ·) is not even the trace of a Radon measure in O(Ω) and thus (1.3) fails. As an example, consider d = 2, N = 1, Ω := (0, 1), v = (v_{1}, v_{2}), and let A(v) = 0 if and only if v′_{2} = 0 as in (ii) above. Let

f(v) := \begin{cases}(v_{1} − 1)^{2} + v_{2}^{2}, & if v_{2} ∈ ℚ \\ (v_{1} + 1)^{2} + v_{2}^{2}, & if v_{2} ∉ ℚ. \end{cases}

Although f satisfies a quadratic growth condition of the type (H), and (A_{3}) holds with q = 2, it is easy to see that for all intervals (a, b) ⊂ (0, 1),

F_{0}((u, v); (a, b)) = F_{0}(v; (a, b)) = min {∫_{a}^{b}((v_{1} − 1)^{2} + v_{2}^{2})dx, ∫_{a}^{b}((v_{1} + 1)^{2} + v_{2}^{2})dx}

which is not the trace of a Radon measure on O(Ω). On the other hand, it may be shown that (see the Appendix below for a proof)

F((u, v); (a, b)) = F(v; (a, b)) = ∫_{a}^{b}(ψ^{∗∗}(v_{1}) + v_{2}^{2}) dx,

where ψ^{∗∗}(v_{1}) is the convex envelope of

ψ(v_{1}) := min {(v_{1} − 1)^{2}, (v_{1} + 1)^{2}} ·

(iv) Using the growth condition (H), a mollification argument, and the linearity of A, it can be shown that (see Rem. 3.3 in [22])

Q_{A}f(x, u, v) = inf { ∫_{Q} f(x, u, v + w(y)) dy : w ∈ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A, ∫_{Q} w(y) dy = 0}·

We write w ∈ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A when w ∈ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) and Aw = 0 in W^{−1,q}(Q; ℝ^{l}).

(v) We may also treat the cases q = 1, ∞ and p = ∞. See Theorem 3.6 below.

The proof of Theorem 1.1 relies heavily on the use of Young measures (see [5, 40]). However, instead of applying directly the arguments of Fonseca and Müller [22] (based on Balder’s [4] and Kristensen’s [26] approach in the curl–free case), we use these together with the blow-up method introduced by Fonseca and Müller in [20].

Although in Theorem 1.1 the functions u and v are not related to each other, the arguments of the proof work equally well when u and v are not independent. Indeed as a corollary, we can prove the following two theorems:

Theorem 1.3. Let 1 ≤ p ≤ ∞, s ∈ ℕ, and suppose that f : Ω × E_{[s−1]}^{n} × E_{s}^{n} → [0, ∞) is a Carathéodory function satisfying

0 ≤ f(x, u, v) ≤ C (1 + |u|^{p} + |v|^{p}) , 1 ≤ p < ∞,

for a.e. x ∈ Ω and all (u, v) ∈ E_{[s−1]}^{n} × E_{s}^{n}, where C > 0, and

f ∈ L_{loc}^{∞}(\overline{Ω} × E_{[s−1]}^{n} × E_{s}^{n}; [0, ∞)) if p = ∞.

Then for every u ∈ W^{s,p}(Ω; ℝ^{n}) we have

∫_{Ω} Q^{s}f(x, u, …, ∇^{s}u) dx = inf { lim inf_{k→∞} ∫_{Ω} f(x, u_{k}, …, ∇^{s}u_{k}) dx : {u_{k}} ⊂ W^{s,p}(Ω; ℝ^{n}),

u_{k} ⇀ u in W^{s,p}(Ω; ℝ^{n}) (⇀^{⋆} if p = ∞)},

where, for a.e. x ∈ Ω and all (u, v) ∈ E_{[s−1]}^{n} × E_{s}^{n},

Q^{s}f(x, u, v) := inf { ∫_{Q} f(x, u, v + ∇^{s}w(y)) dy : w ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{n})}·

Remarks 1.4. (i) Here E_{s}^{n} stands for the space of n-tuples of symmetric s-linear maps on ℝ^{ℕ},

E_{[s−1]}^{n} := ℝ^{n} × E_{1}^{n} × ⋯ × E_{s−1}^{n},

and

∇^{l}u := (\frac{∂^{l}u}{∂x_{1}^{α_{1}} … ∂x_{N}^{α_{N}}} )_{α_{1}+...+α_{N} =l} , l ≥ 1.

(ii) When s = 1 we recover classical relaxation results (see e.g. the work of Acerbi and Fusco [1], Dacorogna [13], Marcellini and Sbordone [28] and the references contained therein).

When s > 1 lower semicontinuity results related to Theorem 1.3 are due to Meyers [29], Fusco [23] and Guidorzi and Poggiolini [25], while we are not aware of any integral representation formula for the relaxed energy, when the integrand depends on the full set of variables, that is f = f(x, u, …, ∇^{s}u). This is due to the fact that classical truncation methods for s = 1 cannot be extended in a simple way to truncate higher order derivatives. The results of Fonseca and Müller (see the proof of Lem. 2.15 in [22]), where the truncation is only on the highest order derivative ∇^{s}u, and Corollary 3.2 below, allows us to overcome this difficulty. Note however that this technique relies heavily on p-equi-integrability, and thus cannot work in the case p = 1, if one replaces weak convergence in W^{s,1}(Ω; ℝ^{n}) with the natural convergence, which is strong convergence in W^{s−1,1}(Ω; ℝ^{n}). In this context, a relaxation result has been given by Amar and De Cicco [2], but only when f = f(∇^{s}u), so that truncation is not needed. The general case where f depends also on lower order derivatives has been addressed by Fonseca et al. [19].

Theorem 1.5. Let 1 ≤ p ≤ ∞, let Ω ⊂ ℝ^{N} be an open, bounded, connected set, and suppose that f : Ω × ℝ^{N} × ℝ^{N^{2}} → [0, ∞) is a Carathéodory function satisfying

0 ≤ f(x, u, v) ≤ C (1 + |u|^{p} + |v|^{p}) , 1 ≤ p < ∞,

for a.e. x ∈ Ω and all (u, v) ∈ ℝ^{N} × ℝ^{N^{2}}, where C > 0, and

f ∈ L_{loc}^{∞}(\overline{Ω} × ℝ^{N} × ℝ^{N^{2}}; [0, ∞)) if p = ∞.

Then for every u ∈ W^{1,p}(Ω; ℝ^{N}) such that div u = 0, we have

∫_{Ω} f̄(x, u(x), ∇u(x)) dx = inf { lim inf_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) dx : {u_{n}} ⊂ W^{1,p}(Ω; ℝ^{N}),

div u_{n} = 0, u_{n} ⇀ u in W^{1,p}(Ω; ℝ^{N}) (⇀^{⋆} if p = ∞)},

(1.7) where, for a.e. x ∈ Ω and all (u, v) ∈ ℝ^{N} × ℝ^{N^{2}},

f̄(x, u, v) := inf { ∫_{Q} f(x, u, v + ∇w(y)) dy : w ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{N}), div w = 0}·

Remark 1.6. To the authors’ knowledge, this result is new in this generality (for a different proof, with additional smoothness assumptions, see [9]). A related problem was addressed by Dal Maso et al. in [16], where it was shown that the Γ-limit of a family of functionals of the type (1.7) may be non local if (H) is violated.

In the second part of the paper we present (Γ-convergence) homogenization results for periodic integrands in the context of A-quasiconvexity. Let ε > 0 and 1 < q < ∞, and consider a family of functionals

F_{ε} : (L^{q}(Ω; ℝ^{d}) ∩ ker A) × O(Ω) → [0, ∞)

defined by

F_{ε}(v; D) := ∫_{D} f (\frac{x}{ε}, v(x)) dx,

where the density f satisfies the following hypotheses:

(A_{1}) f : ℝ^{N} ×ℝ^{d} → [0, ∞) is a continuous function, Q-periodic in the first argument, that is f(x+e_{i}, v) = f(x, v) for every i = 1, …, N, where e_{i} are the elements of the canonical basis of ℝ^{N};

(A_{2}) there exists C > 0 such that

0 ≤ f(x, v) ≤ C(1 + |v|^{q})

for all (x, v) ∈ ℝ^{N} × ℝ^{d};

(A_{3}) there exists C > 0 such that

f(x, v) ≥ \frac{1}{C} |v|^{q} − C

for all (x, v) ∈ ℝ^{N} × ℝ^{d}.

Let ε_{n} → 0^{+}. We say that a functional

J : (L^{q}(Ω; ℝ^{d}) ∩ ker A) × O(Ω) → [0, +∞]

is the Γ − lim inf (resp. Γ − lim sup) of the sequence of functionals {F_{ε_{n}}} with respect to the weak convergence in L^{q}(Ω; ℝ^{d}) if for every v ∈ L^{q}(Ω; ℝ^{d}) ∩ ker A

J (v; Ω) = inf{lim inf_{n→∞} (resp. lim sup_{n→∞}) F_{ε_{n}}(v_{n}; Ω) : v_{n} ∈ L^{q}(Ω; ℝ^{d}) ∩ ker A,
v_{n} ⇀ v in L^{q}(Ω; ℝ^{d})},

(1.8)

and we write

J = Γ − lim inf_{n→∞} F_{ε_{n}} (resp. J = Γ − lim sup_{n→∞} F_{ε_{n}}) .

When finite energy sequences are L^{q}-equibounded then the infimum in the definition of Γ − lim inf (resp. Γ − lim sup) is attained. We say that the sequence {F_{ε_{n}}} Γ-converges to J if the Γ − lim inf and Γ − lim sup coincide, and we write

J = Γ − lim_{n→∞} F_{ε_{n}}.

The functional J is said to be the Γ − lim inf (resp. Γ − lim sup) of the family of functionals {F_{ε}} with respect to the weak convergence in L^{q}(Ω; ℝ^{d}) if for every sequence ε_{n} → 0^{+} we have that

J = Γ − lim inf_{n→∞} F_{ε_{n}} (resp. J = Γ − lim sup_{n→∞} F_{ε_{n}}) ,

and we write

J = Γ − lim inf_{ε→0} F_{ε} (resp. J = Γ − lim sup_{ε→0} F_{ε}) .

Finally, we say that J is the is the Γ-limit of the family of functionals {F_{ε}}, and we write

J = Γ − lim_{n→∞} F_{ε_{n}},

if Γ − lim inf and Γ − lim sup coincide.

In the sequel we will also consider functionals J given by (1.8) where we replace the weak convergence v_{n} ⇀ v with the convergence v_{n} → v with respect to some metric d. In order to highlight this dependence on the metric d these functionals will be denoted as

J = Γ(d) − lim inf_{n→∞} F_{ε_{n}} (resp. J = Γ(d) − lim sup_{n→∞} F_{ε_{n}}) ,

as it is customary (see [10, 15]).

Theorem 1.7. Under hypotheses (A_{1}) − (A_{2}) and the constant-rank hypothesis (1.1),

F_{hom} = Γ − lim inf_{ε→0} F_{ε},

where

F_{hom}(v; D) := ∫_{D} f_{hom}(v) dx

for all v ∈ L^{q}(Ω; ℝ^{d}) ∩ ker A and D ∈ O(Ω), and

f_{hom}(v) := inf_{k∈ℕ} \frac{1}{k^{N}} inf\{∫_{kQ} f(x, v + w(x)) dx : w ∈ L_{k-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A,

∫_{kQ} w(x) dx = 0}

(1.9)

for all v ∈ ℝ^{d}. Moreover, if (A_{3}) holds then

F_{hom} = Γ − lim_{ε→0} F_{ε}.

For the definition of the space L_{k-per}^{q}(ℝ^{N}; ℝ^{d}), we direct the reader to Section 2.

Remarks 1.8. (i) Using the growth condition (A_{2}), a mollification argument, and the linearity of A, it can be shown that

f_{hom}(v) = inf_{k∈ℕ} \frac{1}{k^{N}} inf\{∫_{kQ} f(x, v + w(x)) dx : w ∈ L_{k-per}^{∞}(ℝ^{N}; ℝ^{d}) ∩ ker A,

∫_{kQ} w(x) dx = 0}·

See also Corollary 5.7 below.

(ii) When f satisfies the q-Lipschitz condition

|f(x, v_{1}) − f(x, v_{2})| ≤ C(|v_{1}|^{q−1} + |v_{2}|^{q−1} + 1)|v_{1} − v_{2}|  (1.10)

for all x ∈ ℝ^{N}, v_{1}, v_{2} ∈ ℝ^{d}, and for some C > 0, then the continuity of f(·, v) can be weakened to measurability, namely f can be assumed to be simply Carathéodory. Note that (1.10) is not restrictive when A = curl, that is when v = ∇u for some u ∈ W^{1,q}(Ω; ℝ^{m}), d = N × m. Indeed, in this case in the definition of Γ-convergence we may replace the weak convergence of the gradients in L^{q}(Ω; ℝ^{d}) with the strong convergence in L^{q}(Ω; ℝ^{m}) of the potentials normalized to have zero average over Ω, and thus

Γ − lim_{ε→0} ∫_{Ω} f (\frac{x}{ε}, v(x)) dx = Γ(L^{q}(Ω; ℝ^{m})) − lim_{ε→0} ∫_{Ω} f (\frac{x}{ε}, ∇u(x)) dx

= Γ(L^{q}(Ω; ℝ^{m})) − lim_{ε→0} ∫_{Ω} Qf (\frac{x}{ε}, ∇u(x)) dx,

by Proposition 7.13 in [10]. As shown in [27], if f(x, v) is a Borel function which satisfies the growth condition (A_{2}) then its quasiconvex envelope Qf satisfies (1.10).

A similar argument fails for general A-quasiconvexity, since the function Q_{A}f(x, ·) may not even be continuous, see Remark 1.2(i) above.

In Section 2 we collect preliminary results on Young measures and Γ-convergence. The general relaxation results (see Th. 1.1 and its exstension Th. 3.6) are proved in Section 3, and Section 4 is devoted to the applications of the general relaxation principle to Theorems 1.3 and 1.5. Finally, in Section 5 we address homogenization of functionals of A-constrained vector fields.

2. Preliminaries

We start with some notation. Here Ω is an open, bounded subset of ℝ^{N}, L^{N} is the N dimensional Lebesgue measure, S^{N−1} := {x ∈ ℝ^{N} : |x| = 1} is the unit sphere, and Q := (−1/2, 1/2)^{N} the unit cube centered at the origin. We set Q(x_{0}, ε) := x_{0} +εQ for ε > 0 and x_{0} ∈ ℝ^{N}. A function w ∈ L_{loc}^{q}(ℝ^{N}; ℝ^{d}) is said to be Q–periodic if w(x + e_{i}) = w(x) for a.e. all x ∈ ℝ^{N} and every i = 1, …, N, where (e_{1}, …, e_{N}) is the canonical basis of ℝ^{N}. We write w ∈ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}). More generally, w ∈ L_{loc}^{q}(ℝ^{N}; ℝ^{d}) is said to be kQ–periodic, k ∈ ℕ, if w(k·) is Q–periodic. We write w ∈ L_{k-per}^{q}(ℝ^{N}; ℝ^{d}). Also C_{1-per}^{∞}(ℝ^{N}; ℝ^{d}) will stands for the space of Q–periodic functions in C^{∞}(ℝ^{N}; ℝ^{d}).

We recall briefly some facts about Young measures which will be useful in the sequel (see e.g. [5, 33]). If D is an open set (not necessarily bounded), we denote by C_{c}(D; ℝ^{d}) the set of continuous functions with compact support in D, endowed with the supremum norm. The dual of the closure of C_{c}(D; ℝ^{d}) may be identified with the set of ℝ^{d}-valued Radon measures with finite mass M(D; ℝ^{d}), through the duality

⟨ν, f⟩ := ∫_{D} f(y) dν(y), ν ∈ M(D; ℝ^{d}), f ∈ C_{c}(D; ℝ^{d}).

A map ν : Ω → M(D; ℝ^{d}) is said to be weak-∗ measurable if x ↦ ⟨ν_{x}, f⟩ are measurable for all f ∈ C_{c}(D; ℝ^{d}). The following result is a corollary of the Fundamental Theorem on Young Measures (see [5, 7, 34])

Theorem 2.1. Let z_{n} : Ω → ℝ^{d} be measurable functions such that

sup_{n∈ℕ} ∫_{Ω} |z_{n}|^{q} dx < ∞,

for some q > 0. Then there exists a subsequence {z_{n_{k}}} of {z_{n}} and a weak-∗ measurable map ν : Ω → M(ℝ^{d}; ℝ^{d}) such that (i) ν_{x} ≥ 0, ||ν_{x}||_{M} = ∫_{ℝ^{d}} dν_{x} = 1 for a.e. x ∈ Ω; (ii) if f : Ω × ℝ^{d} → ℝ is a normal function bounded from below then

lim inf_{k→∞} ∫_{Ω} f(x, z_{n_{k}}(x)) dx ≥ ∫_{Ω} \overline{f}(x)dx < ∞,

where

\overline{f}(x) := ⟨ν_{x}, f(x, ·)⟩ = ∫_{ℝ^{d}} f(x, y) dν_{x}(y);

(iii) for any Carathéodory function f : Ω × ℝ^{d} → ℝ bounded from below one has

lim_{k→∞} ∫_{Ω} f(x, z_{n_{k}}(x) dx = ∫_{Ω} \overline{f}(x)dx < ∞

if and only if {f(·, z_{n_{k}}(·))} is equi-integrable.

The map ν : Ω → M(ℝ^{d}; ℝ^{d}) is called theYoung measure generated by the sequence {z_{n_{k}}}.

Proposition 2.2. If {z_{n}} generates a Young measure ν and v_{n} → 0 in measure, then {z_{n} + v_{n}} still generates the Young measure ν.

If 1 < q ≤ ∞ then W^{−1,q}(Ω; ℝ^{l}) is the dual of W_{0}^{1,q′}(Ω; ℝ^{l}), where q′ is the Hölder conjugate exponent of q, that is 1/q + 1/q′ = 1. It is well known that F ∈ W^{−1,q}(Ω; ℝ^{l}) if and only if there exist g_{1}, …, g_{N} ∈ L^{q}(Ω; ℝ^{l}) such that

⟨F, w⟩ = ∑_{i=1}^{N} ∫_{Ω} g_{i} · \frac{∂w}{∂x_{i}} dx for all w ∈ W_{0}^{1,q}(Ω; ℝ^{l}).

Consider a collection of linear operators A^{(i)} : ℝ^{d} → ℝ^{l}, i = 1, …, N, and define the differential operator

\begin{matrix}A : L^{q}(Ω; ℝ^{d}) & ⟶ W^{−1,q}(Ω; ℝ^{l}) \\ v & ⟼ Av \end{matrix}

as follows:

⟨Av, w⟩ := ⟨∑_{i=1}^{N} A^{(i)} \frac{∂v}{∂x_{i}} , w⟩ = − ∑_{i=1}^{N} ∫_{Ω} A^{(i)}v \frac{∂w}{∂x_{i}} dx for all w ∈ W_{0}^{1,q}(Ω; ℝ^{l}).

Even though the operator A so defined depends on Ω, we will omit reference to the underlying domain whenever it is clear from the context. In particular, if v ∈ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) then we will say that v ∈ ker A if Av = 0 in W^{−1,q}(Q; ℝ^{l}).

Throughout the paper we assume that A satisfies the constant-rank property (1.1).

The following proposition is due to Fonseca and Müller [22].

Proposition 2.3. (i) (1 < q < +∞) Let 1 < q < +∞, let {V_{n}} be a bounded sequence in L^{q}(Ω; ℝ^{d}) such that AV_{n} → 0 in W^{−1,q}(Ω; ℝ^{l}), V_{n} ⇀ V in L^{q}(Ω; ℝ^{d}), and assume that {V_{n}} generates a Young measure ν. Then there exists a q-equi-integrable sequence {v_{n}} ⊂ L^{q}(Ω; ℝ^{d}) ∩ ker A such that

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

and, in particular, {v_{n}} still generates ν. Moreover, if Ω = Q then v_{n} − V ∈ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A.

(ii) (q = 1) Let {V_{n}} be a sequence converging weakly in L^{1}(Ω; ℝ^{d}) to a function V, AV_{n} → 0 in W^{−1,r}(Ω; ℝ^{l}) for some r ∈ (1, N/(N − 1)), and assume that {V_{n}} generates a Young measure ν. Then there exists an equi-integrable sequence {v_{n}} ⊂ L^{1}(Ω; ℝ^{d}) ∩ ker A such that

∫_{Ω} v_{n} dx = ∫_{Ω} V dx, ||v_{n} − V_{n}||_{L^{1}(Ω)} → 0,

and, in particular, {v_{n}} still generates ν. Moreover, if Ω = Q then v_{n} − V ∈ L_{1-per}^{1}(ℝ^{N}; ℝ^{d}) ∩ ker A.

(iii) (q = +∞) Let {V_{n}} be a sequence that satisfies V_{n} ⇀^{⋆} V in L^{∞}(Ω; ℝ^{d}), AV_{n} ⇀ 0 in L^{r}(Ω) for some r > N, and assume that {V_{n}} generates a Young measure ν. Then there exists a sequence {v_{n}} ⊂ L^{∞}(Ω; ℝ^{d}) ∩ ker A such that

∫_{Ω} v_{n} dx = ∫_{Ω} V dx, ||v_{n} − V_{n}||_{L^{∞}(Ω)} → 0,

and, in particular, {v_{n}} still generates ν. Moreover, if Ω = Q then v_{n} − V ∈ L_{1-per}^{∞}(ℝ^{N}; ℝ^{d}) ∩ ker A.

In the second part of the paper we will need the following classical results from Γ-convergence. For a proof see [10].

Proposition 2.4. Let (X, d) be a separable metric space and let f_{n} : X → [−∞, ∞]. Then (i) there exists an increasing sequence of integers {n_{k}} such that

Γ(d) − lim_{k→∞} f_{n_{k}}(x) exists for all x ∈ X.

(ii) Moreover

f_{∞} = Γ(d) − lim_{n→∞} f_{n}

if and only if for every subsequence {f_{n_{k}}} there exists a further subsequence {f_{n_{k_{j}}} } which Γ(d)-converges to f_{∞}.

3. Relaxation

In this section we prove Theorem 1.1 and its generalization to the case where q ∈ {1, ∞} and p = ∞ (see Th. 3.6).

Lemma 3.1. Let f : Ω × ℝ^{m} × ℝ^{d} → [0, ∞) be a Carathéodory function satisfying (H), with 1 ≤ p < ∞ and 1 < q < ∞. Let (u, v) ∈ L^{p}(D; ℝ^{m}) × (L^{q}(D; ℝ^{d}) ∩ ker A), where D ∈ O(Ω), and consider a sequence of functions {(u_{k}, v̂_{k})} ⊂ L^{p}(D; ℝ^{m}) × L^{q}(D; ℝ^{d}) such that

\begin{matrix}u_{k} → u & in L^{p}(D; ℝ^{m}), v̂_{k} ⇀ v in L^{q}(D; ℝ^{d}) \\ Av̂_{k} → 0 in W^{−1,q}(D; ℝ^{l}). \end{matrix}  (3.1)

Then we can find a q-equi-integrable sequence {v_{k}} ⊂ L^{q}(D; ℝ^{d}) ∩ ker A such that

v_{k} ⇀ v in L^{q}(D; ℝ^{d}), ∫_{D} v_{k} dx = ∫_{D} v dx,

and

lim inf_{k→∞} ∫_{D} f(x, u(x), v_{k}(x)) dx ≤ lim inf_{k→∞} ∫_{D} f(x, u_{k}(x), v̂_{k}(x)) dx.

Proof of Lemma 3.1. Consider a subsequence {(u_{n}, v̂_{n})} of {(u_{k}, v̂_{k})} such that

lim_{n→∞} ∫_{D} f(x, u_{n}(x), v̂_{n}(x)) dx = lim inf_{k→∞} ∫_{D} f(x, u_{k}(x), v̂_{k}(x)) dx

and {(u_{n}, v̂_{n})} generates the Young measure {δ_{u(x)} ⊗ ν_{x}}_{x∈D}. For i ∈ ℕ let

F_{i} := \{x ∈ D : dist(x, ∂D) < \frac{1}{i}\} ,

and consider cut-off functions θ_{i} with compact support in D and such that θ_{i} ≡ 1 in D \ F_{i}. Set w_{i,n} := θ_{i}(v̂_{n} − v) ∈ L^{q}(D; ℝ^{d}) and fix φ ∈ L^{q′}(D; ℝ^{d}), where q′ is the Hölder conjugate exponent of q. Then

lim_{i→∞} lim_{n→∞} ∫_{D} φ(x)w_{i,n}(x) dx = lim_{i→∞} lim_{n→∞} ∫_{D} φ(x)θ_{i}(x)(v̂_{n}(x) − v(x))dx = 0,  (3.2)

where we have used the fact that v̂_{k} ⇀ v in L^{q}(D; ℝ^{d}). Hence w_{i,n} ⇀ 0 in L^{q}(D; ℝ^{d}) as n → ∞ and i → ∞. Moreover, in view of the compact embedding

L^{q}(D; ℝ^{l}) ↪ W^{−1,q}(D; ℝ^{l})

and the assumption that Av̂_{k} → 0 in W^{−1,q}(D; ℝ^{l}), we have that

lim_{i→∞} lim_{n→∞} Aw_{i,n} = 0 in W^{−1,q}(D; ℝ^{l}).

Let 𝔾 be a countable dense subset of L^{q}(D; ℝ^{d}). By means of a diagonalization process we obtain subsequences {u_{i} := u_{n_{i}}} and {ŵ_{i} := w_{i,n_{i}} = θ_{i}(v̂_{n_{i}} − v)} such that ||u_{i} − u||_{L^{p}} → 0, (3.2) holds for each φ ∈ 𝔾, and

Aŵ_{i} → 0 in W^{−1,q}(D; ℝ^{l}).

Hence ŵ_{i} ⇀ 0 in L^{q}(D; ℝ^{d}), by the density of 𝔾 in L^{q}(D; ℝ^{d}). By Proposition 2.3(i) there exists a q-equi-integrable sequence {w_{i}} ⊂ L^{q}(D; ℝ^{d}) ∩ ker A such that w_{i} ⇀ 0 in L^{q}(D; ℝ^{d}), and

∫_{D} w_{i} dx = 0, ||ŵ_{i} − w_{i}||_{L^{s}(D)} → 0 for all 1 ≤ s < q.  (3.3)

Set v_{i} := v + w_{i}. Then ∫_{D} v_{i} dx = ∫_{D} v dx, v_{i} ⇀ v in L^{q}(D; ℝ^{d}). By Hölder’s inequality and by (3.3), for 1 ≤ s < q

\begin{aligned}||v̂_{n_{i}} − v_{i}||_{L^{s}(D)} & ≤ ||v̂_{n_{i}} − v − ŵ_{i}||_{L^{s}(D)} + ||ŵ_{i} − w_{i}||_{L^{s}(D)} \\ & ≤ ||(1 − θ_{i})(v̂_{n_{i}} − v)||_{L^{s}(D)} + ||ŵ_{i} − w_{i}||_{L^{s}(D)} \\ & ≤ ||v̂_{n_{i}} − v||_{L^{q}(D)}|F_{i}|^{r} + ||ŵ_{i} − w_{i}||_{L^{s}(D)} → 0 \end{aligned}  (3.4)

as i → ∞ and where r := (q − s)/sq. By (3.4) and Proposition 2.2, the two sequences

{(u(x), v_{i}(x))}  and  {(u_{i}(x), v̂_{i}(x))}

generate the same Young measure {δ_{u(x)} ⊗ ν_{x}}_{x∈D}. Hence by Theorems 2.1(ii) and (iii)

lim_{i→∞} ∫_{D} f(x, u(x), v_{i}(x)) dx = ∫_{D} ∫_{ℝ^{d}} f(x, u(x), V)dν_{x}(V)dx ≤ lim inf_{i→∞} ∫_{D} f(x, u_{i}(x), v̂_{i}(x)) dx

= lim inf_{k→∞} ∫_{D} f(x, u_{k}(x), v̂_{k}(x)) dx,

(3.5) where we have used the fact that {f(x, u(x), v_{i}(x))} is equi-integrable over D, which follows from (H) and the q-equi-integrability of {v_{i}} over D.

It follows immediately from Lemma 3.1 that under its assumptions on f it holds:

Corollary 3.2. For D ∈ O(Ω) and (u, v) ∈ L^{p}(Ω; ℝ^{m}) × (L^{q}(Ω; ℝ^{d}) ∩ ker A)

F((u, v); D) = inf{lim inf_{n→∞} ∫_{D} g(x, v_{n}(x)) dx : {v_{n}} ⊂ L^{q}(D; ℝ^{d}) ∩ ker A is q-equi-integrable
and v_{n} ⇀ v in L^{q}(D; ℝ^{d})},

where g is the Carathéodory function defined by

g(x, v) := f(x, u(x), v).

Note that, by (H), the function g satisfies the growth condition

0 ≤ g(x, v) ≤ C (1 + |u(x)|^{p} + |v|^{q})  (3.6)

for a.e. x ∈ Ω and all v ∈ ℝ^{d}. Moreover, since g is a Carathéodory function, by the Scorza-Dragoni theorem for each j ∈ ℕ there exists a compact set K_{j} ⊂ Ω, with |Ω \ K_{j}| ≤ 1/j, such that g : K_{j} × ℝ^{d} → [0, ∞) is continuous. Let K_{j}^{∗} be the set of Lebesgue points of χ_{K_{j}}, and set

ω := ⋃_{j}^{∞}(K_{j} ∩ K_{j}^{∗}) ∩ L(u, v),  (3.7)

where L(u, v) is the set of Lebesgue points of (u, v). Then

|Ω \ ω| ≤ |Ω \ K_{j}| ≤ \frac{1}{j} → 0 as j → ∞.

Corollary 3.3. Assume that x_{0} ∈ ω, let v ∈ L^{q}(Q; ℝ^{d}) ∩ ker A, and consider r_{k} → 0^{+} and a sequence of functions

{v̂_{k}} ⊂ L^{q}(Q; ℝ^{d}) ∩ ker A

such that

v̂_{k} ⇀ v in L^{q}(Q; ℝ^{d}).

Then we can find a q-equi-integrable sequence {w_{k}} ⊂ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A such that

w_{k} ⇀ 0 in L^{q}(Q; ℝ^{d}), ∫_{Q} w_{k} dx = 0,

and

lim inf_{k→∞} ∫_{Q} g(x_{0}, v(y) + w_{k}(y)) dy ≤ lim inf_{k→∞} ∫_{Q} g(x_{0} + r_{k}y, v̂_{k}(y)) dy.

Proof of Corollary 3.3. We proceed as in the proof of Lemma 3.1 up to (3.4). Since the sequence {v_{i}} is q-equi-integrable, for any η > 0 there exists δ > 0 such that

sup_{i} ∫_{D} C(1 + |u(x_{0})|^{p} + |v_{i}(y)|^{q}) dy < η  (3.8)

for any measurable set D ⊂ Q, with |D| < δ, and where C is the constant given in (H). Fix η > 0 and let δ > 0 be given according to (3.8). By the Biting Lemma (see [6]) we may find a further subsequence {v̂_{n_{j}}} ⊂ {v̂_{n_{i}}} and a set E ⊂ Q such that |Q \ E| < δ and {v̂_{n_{j}}} is q-equi-integrable over E. Hence there exists 0 < δ_{1} < δ such that

sup_{j} ∫_{D} C(1 + |u(x_{0})|^{p} + |v̂_{n_{j}}(y)|^{q}) dy < η  (3.9)

for any measurable set D ⊂ E, with |D| < δ_{1}. Moreover, as {v̂_{n_{j}}}, {v_{j}} are bounded in L^{q}(Q; ℝ^{d}), we may find L > 0 such that

|E \ E_{j}| ≤ δ_{1}, where E_{j} := {y ∈ E : |v̂_{n_{j}}(y)| ≤ L, |v_{j}(y)| ≤ L} ·  (3.10)

Note that by construction of v_{i} and by Proposition 2.3, v_{i} = v + w_{i} where w_{i} ∈ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A. From the definition of the set ω there exists an integer j_{0} such that x_{0} ∈ K_{j_{0}} ∩ K_{j_{0}}^{∗}. Since

g : K_{j_{0}} × \overline{B_{d}(0, L)} → [0, ∞)

is uniformly continuous, there exists ρ > 0 such that

|g(x, v) − g(x_{1}, v)| ≤ η  (3.11)

for all (x, v), (x_{1}, v) ∈ K_{j_{0}} × \overline{B_{d}(0, L)}, with |x − x_{1}| ≤ ρ. By (3.10) and (3.11)

lim_{i→∞} ∫_{Q} g(x_{0} + r_{n_{i}}y, v̂_{n_{i}}(y)) dy ≥ lim inf_{j→∞} ∫_{E_{j}} g(x_{0} + r_{n_{j}}y, v̂_{n_{j}}(y)) dy

≥ lim inf_{j→∞} \frac{1}{r_{n_{j}}^{N}} ∫_{(x_{0}+r_{n_{j}} E_{j})∩K_{j_{0}}} g(x, v̂_{n_{j}}((x − x_{0})/r_{n_{j}})) dx

≥ −η + lim inf_{j→∞} \frac{1}{r_{n_{j}}^{N}} ∫_{(x_{0}+r_{n_{j}} E_{j})∩K_{j_{0}}} g(x_{0}, v̂_{n_{j}}((x − x_{0})/r_{n_{j}})) dx.

(3.12)

Using, once again, the fact that |v̂_{n_{j}}(y)| ≤ L for y ∈ E_{j}, by (3.6) we have that

\frac{1}{r_{n_{j}}^{N}} ∫_{(x_{0}+r_{n_{j}} E_{j})\K_{j_{0}}} g(x_{0}, v̂_{n_{j}}((x − x_{0})/r_{n_{j}})) dx ≤ C(1 + |u(x_{0})|^{p} + L^{q})\frac{|Q(x_{0}, r_{n_{j}}) \ K_{j_{0}}|}{r_{n_{j}}^{N}} → 0

as j → ∞, because x_{0} is a Lebesgue point of χ_{K_{j_{0}}} . Consequently, from (3.12) we get

lim_{i→∞} ∫_{Q} g(x_{0} + r_{n_{i}}y, v̂_{n_{i}}(y)) dy ≥ −η + lim inf_{j→∞} \frac{1}{r_{n_{j}}^{N}} ∫_{x_{0}+r_{n_{j}} E_{j}} g(x_{0}, v̂_{n_{j}}((x − x_{0})/r_{n_{j}})) dx

= −η + lim inf_{j→∞} ∫_{E_{j}} g(x_{0}, v̂_{n_{j}}(y)) dy

≥ −2η + lim inf_{j→∞} ∫_{E} g(x_{0}, v̂_{n_{j}}(y)) dy,

where we have used (3.6, 3.9) and the fact that |E \ E_{j}| ≤ δ_{1}. We may now proceed as in the previous lemma, using the Carathéory function h(x, v) := χ_{E}(x)g(x_{0}, v), to obtain

lim_{i→∞} ∫_{Q} g(x_{0} + r_{n_{i}}y, v̂_{n_{i}}(y)) dy ≥ −2η + lim inf_{j→∞} ∫_{E} g(x_{0}, v_{j}(y)) dy ≥ −3η + lim inf_{j→∞} ∫_{Q} g(x_{0}, v_{j}(y)) dy

by (3.8). It now suffices to let η → 0^{+}.

Theorem 1.1 follows from Lemmas 3.4 and 3.5 below. We will use the notation μ⌊A to denote the restriction of a Radon measure μ to the Borel set A, i.e., μ⌊A(X) := μ(X ∩ A) where X is an arbitrary Borel set in the domain of μ.

Lemma 3.4. F((u, v); ·) is the trace of a Radon measure absolutely continuous with respect to L^{N}⌊Ω.

Proof of Lemma 3.4. As it is usual, it suffices to prove subadditivity (see e.g. [3, 21]), i.e.

F((u, v); D) ≤ F((u, v); D \ \overline{B}) + F((u, v); C)

if B ⊂⊂ C ⊂⊂ D. Fix η > 0. By Corollary 3.2 there exist two q-equi-integrable sequences

\{v_{k}\} ⊂ L^{q}(D \ \overline{B}; ℝ^{d}) ∩ ker A, \{w_{k}\} ⊂ L^{q}(C; ℝ^{d}) ∩ ker A,

such that

v_{k} ⇀ v in L^{q}(D \ \overline{B}; ℝ^{d}), w_{k} ⇀ v in L^{q}(C; ℝ^{d}),

and

\begin{matrix}lim_{k→∞} & ∫_{D\\overline{B}} g(x, v_{k}(x)) dx ≤ F((u, v); D \ \overline{B}) + η, \\ lim_{k→∞} & ∫_{C} g(x, w_{k}(x)) dx ≤ F((u, v); C) + η. \end{matrix}

Let θ_{j} be smooth cut-off functions, θ_{j} ∈ C_{c}^{∞}(C; [0, 1]), θ_{j}(x) = 1 for all x ∈ B, and |{0 < θ_{j} < 1}| → 0 as j → ∞. Set

V̂_{j,k} := (1 − θ_{j})v_{k} + θ_{j}w_{k}.

Then, for j fixed,

AV̂_{j,k} = (1 − θ_{j})Av_{k} + θ_{j}Aw_{k} − ∑_{i=1}^{N} A^{(i)}v_{k} \frac{∂θ_{j}}{∂x_{i}} + ∑_{i=1}^{N} A^{(i)}w_{k} \frac{∂θ_{j}}{∂x_{i}} → 0

as k → ∞ in W^{−1,q}(D; ℝ^{l}) strong. Using a diagonalization procedure such as that adopted in the proof of Lemma 3.1, we get

V̂_{j} ⇀ v in L^{q}(D; ℝ^{d}), AV̂_{j} → 0 in W^{−1,q}(D; ℝ^{l}),

where V̂_{j} := V̂_{j,k_{j}}. By Lemma 3.1 we can find a q-equi-integrable sequence {V_{j}} ⊂ L^{q}(D; ℝ^{d}) ∩ ker A such that V_{j} ⇀ v in L^{q}(D; ℝ^{d}) and

lim inf_{j→∞} ∫_{D} g(x, V_{j}(x)) dx ≤ lim inf_{j→∞} ∫_{D} g(x, V̂_{j}(x)) dx.

16 A. BRAIDES, I. FONSECA AND G. LEONI

Consequently, in view of Corollary 3.2

F((u, v); D) ≤ lim inf_{j→∞} ∫_{D} g(x, V_{j}(x)) dx ≤ lim inf_{j→∞} ∫_{D} g(x, V̂_{j}(x)) dx

≤ lim sup_{j→∞} ∫_{{θ_{j}=0}} g(x, v_{k_{j}}(x)) dx + lim sup_{j→∞} ∫_{{θ_{j}=1}} g(x, w_{k_{j}}(x)) dx

+ lim sup_{j→∞} ∫_{{0<θ_{j}<1}} C(1 + |u(x)|^{p} + |w_{k_{j}}(x)|^{q} + |v_{k_{j}}(x)|^{q}) dx

≤ 2η + F((u, v); D \ \overline{B}) + F((u, v); C).

It suffices to let η → 0^{+}. Finally, note that by (H) we have that F((u, v), ·) ≤ C(1 + |u|^{p} + |v|^{q})L^{N}⌊Ω.

Lemma 3.5. For L^{N} a.e. x_{0} ∈ Ω we have

\frac{dF((u, v); ·)}{dL^{N}}(x_{0}) = Q_{A}f(x_{0}, u(x_{0}), v(x_{0})).

Proof of Lemma 3.5. Fix x_{0} ∈ ω, where ω is defined as in (3.7), and such that

lim_{r→0^{+}} \frac{1}{r^{N}} ∫_{Q(x_{0},r)} |u(x) − u(x_{0})|^{p} dx = lim_{r→0^{+}} \frac{1}{r^{N}} ∫_{Q(x_{0},r)} |v(x) − v(x_{0})|^{q} dx = 0  (3.13)

and

\frac{dF((u, v); ·)}{dL^{N}}(x_{0}) = lim_{r→0^{+}} \frac{F((u, v); Q(x_{0}, r))}{r^{N}} < ∞,

where, by virtue of Lemma 3.4, we have chosen the radii r → 0^{+} such that

F((u, v); ∂(Q(x_{0}, r))) = 0.

By Corollary 3.2 and for r > 0 fixed, let {v_{n,r}} ⊂ L^{q}(Q(x_{0}, r); ℝ^{d}) ∩ ker A be such that v_{n,r} ⇀ v in L^{q}(Q(x_{0}, r); ℝ^{d}) as n → ∞ and

lim_{n→∞} ∫_{Q(x_{0},r)} g(x, v_{n,r}(x)) dx ≤ F((u, v); Q(x_{0}, r)) + r^{N+1}.

Then

\frac{dF((u, v); ·)}{dL^{N}}(x_{0}) ≥ lim inf_{r→0^{+}} lim_{n→∞} \frac{1}{r^{N}} ∫_{Q(x_{0},r)} g(x, v_{n.r}(x)) dx = lim inf_{r→0^{+}} lim_{n→∞} ∫_{Q} g(x_{0} + ry, v(x_{0}) + w_{n,r}(y)) dy

where w_{n,r}(y) := v_{n,r}(x_{0} + ry) − v(x_{0}). We claim that w_{n,r} ⇀ 0 in L^{q}(Q; ℝ^{d}) if we first let n → ∞ and then r → 0^{+}. Indeed let φ ∈ L^{q′}(Q; ℝ^{d}), where q′ is the Hölder conjugate exponent of q. Using Hölder’s inequality and then making a change of variables, we get

|∫_{Q} φ(y) w_{r,n}(y) dy| ≤ |∫_{Q} φ(y)(v_{n,r}(x_{0} + ry) − v(x_{0} + ry)) dy| + |∫_{Q} φ(y)(v(x_{0} + ry) − v(x_{0})) dy|

≤ |\frac{1}{r^{N}} ∫_{Q(x_{0},r)} φ((x − x_{0})/r)(v_{n,r}(x) − v(x)) dx|

+ ||φ||_{L^{q′} (Q)} (\frac{1}{r^{N}} ∫_{Q(x_{0},r)} |v(x) − v(x_{0})|^{q} dx)^{1/q} .

If we now let n → ∞ the first integral tends to zero, since v_{n,r} ⇀ v in L^{q}(Q(x_{0}, r); ℝ^{d}). The claim then follows by letting r → 0^{+} and by using (3.13). Diagonalize to get ŵ_{k} ∈ L^{q}(Q; ℝ^{d})∩ker A such that ŵ_{k} ⇀ 0 in L^{q}(Q; ℝ^{d}) and

\frac{dF((u, v); ·)}{dL^{N}}(x_{0}) ≥ lim_{k→∞} ∫_{Q} g(x_{0} + r_{k} y, v(x_{0}) + ŵ_{k}(y)) dy

where r_{k} → 0. By Corollary 3.3 there is a q-equi-integrable sequence {w_{k}} ⊂ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A such that

w_{k} ⇀ 0 in L^{q}(Q; ℝ^{d}), ∫_{Q} w_{k} dy = 0,

and

\frac{dF((u, v); ·)}{dL^{N}}(x_{0}) ≥ lim_{k→∞} ∫_{Q} g(x_{0} + r_{k} y, v(x_{0}) + ŵ_{k}(y)) dy

≥ lim inf_{k→∞} ∫_{Q} f(x_{0}, u(x_{0}), v(x_{0}) + w_{k}(y)) dy ≥ Q_{A}f(x_{0}, u(x_{0}), v(x_{0})).

To conclude the proof of the lemma it remains to show that

\frac{dF((u, v); ·)}{dL^{N}}(x_{0}) ≤ Q_{A}f(x_{0}, u(x_{0}), v(x_{0})) for L^{N} a.e. x_{0} ∈ Ω.

Fix η > 0 and let w ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{d}) ∩ ker A be such that ∫_{Q} w dy = 0 and

∫_{Q} f(x_{0}, u(x_{0}), v(x_{0}) + w(y)) dy ≤ Q_{A}f(x_{0}, u(x_{0}), v(x_{0})) + η.  (3.14)

For any fixed r > 0 set w_{n,r}(x) := w(n(x − x_{0})/r). Then w_{n,r} ⇀ 0 in L^{q}(Q(x_{0}, r); ℝ^{d}) as n → ∞. Hence, by Corollary 3.2,

\frac{dF((u, v); ·)}{dL^{N}}(x_{0}) = lim_{r→0^{+}} \frac{F((u, v); Q(x_{0}, r))}{r^{N}} ≤ lim inf_{r→0^{+}} lim inf_{n→∞} \frac{1}{r^{N}} ∫_{Q(x_{0},r)} g(x, v(x) + w_{n,r}(x)) dx.  (3.15)

Fix L > |v(x_{0})| + ||w||_{L^{∞}} + 1, and let j be such that x_{0} ∈ K_{j} ∩ K_{j}^{∗} ∩ L(u, v), where we are using the notation introduced in (3.7). Since

g : K_{j} × \overline{B_{d}(0, L)} → [0, ∞)

is uniformly continuous, there exists 0 < ρ < 1 such that

|g(x, v) − g(x_{1}, v_{1})| ≤ η  (3.16)

for all (x, v), (x_{1}, v_{1}) ∈ K_{j} × \overline{B_{d}(v(x_{0}), L)}, with |x − x_{1}| ≤ ρ and |v − v_{1}| ≤ ρ. Let

E_{r,ρ} := {x ∈ Q(x_{0}, r) : |v(x) − v(x_{0})| ≤ ρ}·

We claim that

lim sup_{r→0^{+}} \frac{1}{r^{N}} ∫_{Q(x_{0},r)\(E_{r,ρ}∩K_{j})} C(1 + |u(x)|^{p} + |v(x)|^{q} + ||w||_{L^{∞}}^{q})dx = 0.  (3.17)

Since |v(x) − v(x_{0})| ≥ ρ for x ∈ Q(x_{0}, r) \ E_{r,ρ}, we have

\begin{aligned}\frac{1}{r^{N}} ∫_{Q(x_{0},r)\(E_{r,ρ}∩K_{j})} C(1 + |u(x)|^{p} + |v(x)|^{q} & + ||w||_{L^{∞}}^{q})dx ≤ C \frac{|Q(x_{0}, r) \ (E_{r,ρ} ∩ K_{j})|}{r^{N}} \\ & + \frac{C}{r^{N}} ∫_{Q(x_{0},r)} (|u(x) − u(x_{0})|^{p} + |v(x) − v(x_{0})|^{q}) dx. \end{aligned}

and

\begin{aligned}\frac{|Q(x_{0}, r) \ (E_{r,ρ} ∩ K_{j})|}{r^{N}} ≤ \frac{|Q(x_{0}, r) \ K_{j}|}{r^{N}} & + \frac{|Q(x_{0}, r) \ E_{r,ρ}|}{r^{N}} ≤ \frac{|Q(x_{0}, r) \ K_{j}|}{r^{N}} \\ & + \frac{1}{ρ^{q}} \frac{C}{r^{N}} ∫_{Q(x_{0},r)} |v(x) − v(x_{0})|^{q} dx → 0 as r → 0^{+}, \end{aligned}

where we have used (3.13) and the fact that x_{0} is a Lebesgue point of χ_{K_{j}}. Then by (3.6, 3.15–3.17) and (3.14),

\frac{dF((u, v); ·)}{dL^{N}}(x_{0}) ≤ lim inf_{r→0^{+}} lim inf_{n→∞} \frac{1}{r^{N}} ∫_{E_{r,ρ}∩K_{j}} g(x, v(x) + w_{n,r}(x)) dx

+ lim sup_{r→0^{+}} \frac{1}{r^{N}} ∫_{Q(x_{0},r)\(E_{r,ρ}∩K_{j})} C(1 + |u(x)|^{p} + |v(x)|^{q} + ||w||_{L^{∞}}^{q})dx

≤ η + lim inf_{r→0^{+}} lim inf_{n→∞} \frac{1}{r^{N}} ∫_{Q(x_{0},r)} g(x_{0}, v(x_{0}) + w_{n,r}(x)) dx

= η + lim inf_{n→∞} ∫_{Q} g(x_{0}, v(x_{0}) + w(ny)) dy

= η + ∫_{Q} g(x_{0}, v(x_{0}) + w(y)) dy ≤ 2η + Q_{A}f(x_{0}, u(x_{0}), v(x_{0})),

by virtue of the equality

lim inf_{n→∞} ∫_{Q} g(x_{0}, v(x_{0}) + w(ny)) dx = ∫_{Q} g(x_{0}, v(x_{0}) + w(y)) dy,

which follows from the Q-periodicity of the function g(x_{0}, v(x_{0}) + w(·)). It now suffices to let η → 0^{+}.

As mentioned in the Introduction, Theorem 1.1 continues to hold when q ∈ {1, ∞} and p = ∞. Indeed, let 1 ≤ p, q ≤ ∞ and assume that

(A_{4}) f : Ω × ℝ^{m} × ℝ^{d} → [0, ∞) is a Carathéodory function satisfying the following growth conditions for a.e. x ∈ Ω and all (u, v) ∈ ℝ^{m} × ℝ^{d}:

0 ≤ f(x, u, v) ≤ C (1 + |u|^{p} + |v|^{q})  if 1 ≤ p, q < ∞,  (3.18)

where C > 0;

0 ≤ f(x, u, v) ≤ a(x, u) (1 + |v|^{q}) if p = ∞ and 1 ≤ q < ∞,

where a ∈ L_{loc}^{∞}(\overline{Ω} × ℝ^{d}; [0, ∞));

0 ≤ f(x, u, v) ≤ b(x, v) (1 + |u|^{p}) if 1 ≤ p < ∞ and q = ∞,

where b ∈ L_{loc}^{∞}(\overline{Ω} × ℝ^{m}; [0, ∞));

f ∈ L_{loc}^{∞}(\overline{Ω} × ℝ^{m} × ℝ^{d}; [0, ∞)) if p = q = ∞.

For D ∈ O(Ω) and (u, v) ∈ L^{p}(Ω; ℝ^{m}) × (L^{q}(Ω; ℝ^{d}) ∩ ker A) define

F((u, v); D) := inf{ lim inf_{n→∞} F ((u_{n}, v_{n}); D) : (u_{n}, v_{n}) ∈ L^{p}(D; ℝ^{m}) × L^{1}(D; ℝ^{d}),
u_{n} → u in L^{p}(D; ℝ^{m}), v_{n} ⇀ v in L^{1}(D; ℝ^{d}), Av_{n} → 0 in W^{−1,r}(D; ℝ^{l})}

if q = 1 and for some r ∈ (1, N/(N − 1)); as in (1.2), we set

F((u, v); D) := inf{ lim inf_{n→∞} F ((u_{n}, v_{n}); D) : (u_{n}, v_{n}) ∈ L^{p}(D; ℝ^{m}) × L^{q}(D; ℝ^{d})
u_{n} → u in L^{p}(D; ℝ^{m}), v_{n} ⇀ v in L^{q}(D; ℝ^{d}), Av_{n} → 0 in W^{−1,q}(D; ℝ^{l}) }

if 1 < q < ∞;

F((u, v); D) := inf{ lim inf_{n→∞} F ((u_{n}, v_{n}); D) : (u_{n}, v_{n}) ∈ L^{p}(D; ℝ^{m}) × L^{∞}(D; ℝ^{d}),
u_{n} → u in L^{p}(D; ℝ^{m}), v_{n} ⇀^{⋆} v in L^{∞}(D; ℝ^{d}), Av_{n} ⇀ 0 in L^{r}(D; ℝ^{l})}

if q = ∞ and for some r > N.

We can prove the following theorem:

Theorem 3.6. Under condition (A_{4}) and the constant-rank hypothesis (1.1), for all D ∈ O(Ω), u ∈ L^{p}(Ω; ℝ^{m}) and v ∈ L^{q}(Ω; ℝ^{d}) ∩ ker A, we have

F((u, v); D) = ∫_{D} Q_{A}f(x, u(x), v(x)) dx.

Proof of Theorem 3.6. Step 1. Assume first that 1 ≤ p < ∞ and q = 1. The proof is similar to the one of Theorem 1.1, with the exceptions that in Lemma 3.1 condition (3.1) should be replaced by

u_{k} → u in L^{p}(D; ℝ^{m}), v̂_{k} ⇀ v in L^{1}(D; ℝ^{d}),
Av̂_{n} → 0 in W^{−1,r}(D; ℝ^{l}) for some r ∈ (1, N/(N − 1)),

that we use the compact embedding

L^{1}(D; ℝ^{l}) ↪ W^{−1,r}(D; ℝ^{l}), r ∈ (0, N/(N − 1),

to diagonalize {w_{i,n}}, and (3.3, 3.4) are replaced, respectively, by

∫_{D} w_{i} dx = 0, ||ŵ_{i} − w_{i}||_{L^{1}(D)} → 0,

\begin{aligned}||v̂_{n_{i}} − v_{i}||_{L^{1}(D)} & ≤ ||v̂_{n_{i}} − v − ŵ_{i}||_{L^{1}(D)} + ||ŵ_{i} − w_{i}||_{L^{1}(D)} ≤ ||(1 − θ_{i})(v̂_{n_{i}} − v)||_{L^{1}(D)} + ||ŵ_{i} − w_{i}||_{L^{1}(D)} \\ & ≤ ||v̂_{n_{i}} − v||_{L^{1}(F_{i})} + ||ŵ_{i} − w_{i}||_{L^{1}(D)} → 0, \end{aligned}

where we have used the fact that ||v̂_{n_{i}} − v||_{L^{1}(F_{i})} → 0 as i → ∞, which is due to the equi-integrability of the original sequence {v̂_{k} − v} and the fact that |F_{i}| → 0.

Step 2. If p = ∞ and 1 ≤ q < ∞ then in Lemma 3.1 the only change needed is in deriving (3.5), which now follows from the fact that, by (3.18),

0 ≤ f(x, u(x), v_{i}(x)) ≤ A_{∞} (1 + |v_{i}(x)|^{q}) ,

where A_{∞} := sup{a(x, u) : x ∈ Ω, |u| ≤ ||u||_{∞}} < ∞, and thus equi-integrability of {f(x, u, v_{i})} follows from the q-equi-integrability of {v_{i}} over D. Moreover in the remaining of the proof of Theorem 1.1, the growth condition (3.6) should be replaced by

0 ≤ g(x, v) ≤ A_{∞} (1 + |v|^{q})  (3.19)

for a.e. x ∈ Ω and all v ∈ ℝ^{d}.

Step 3. If 1 ≤ p ≤ ∞ and q = ∞ then in Lemma 3.1 the hypothesis (3.1) should be replaced by

u_{k} → u in L^{p}(D; ℝ^{m}), v̂_{k} ⇀^{⋆} v in L^{∞}(D; ℝ^{d}),
Av̂_{n} ⇀ 0 in L^{r}(D; ℝ^{l}) for some r > N,

the growth condition should be replaced by (3.19) if 1 ≤ p < ∞, q = ∞, and by g ∈ L_{loc}^{∞}(\overline{Ω} × ℝ^{d}; [0, ∞)) if p = q = ∞, and we can proceed similarly to the proof of Lemma 3.1 to show that w_{i,n} ⇀^{⋆} 0 in L^{∞}(D; ℝ^{d}) and Aw_{i,n} ⇀ 0 in L^{r}(D; ℝ^{l}), and use Proposition 2.3(iii) to get

||v_{i} − v̂_{n_{i}}||_{∞} → 0.

We omit the details.

4. Proofs of Theorems 1.3 and 1.5

Proof of Theorem 1.3. We present the proof for 1 ≤ p < ∞, the case p = ∞ being very similar. Fix u ∈ W^{s,p}(Ω; ℝ^{n}), and for D ∈ O(Ω) define

F(u; D) := inf { lim inf_{k→∞} ∫_{D} f(x, u_{k}, …, ∇^{s}u_{k}) dx : {u_{k}} ⊂ W^{s,p}(D; ℝ^{n}),

u_{k} ⇀ u in W^{s,p}(D; ℝ^{n})},

and let g be the Carathéodory function

g(x, v) := f(x, u(x), …, ∇^{s−1}u(x), v).

Reasoning as in Lemma 3.4, it is easy to show that F(u; ·) is the trace of a Radon measure absolutely continuous with respect to L^{N}⌊Ω.

For any function v ∈ L^{p}(Ω; E_{s}^{n}) set

G(v; D) := inf{lim inf_{k→∞} ∫_{D} g(x, V_{k}(x)) dx : {V_{k}} ⊂ L^{p}(D; E_{s}^{n}) ∩ ker A is p-equi-integrable,
and V_{k} ⇀ v in L^{p}(D; E_{s}^{n})},

where the differential operator A is given by

Av := (\frac{∂}{∂x_{i}} v_{i_{1}...i_{h}ji_{h+2}...i_{s}} − \frac{∂}{∂x_{j}} v_{i_{1}...i_{h}ii_{h+2}...i_{s}})_{0≤h≤s−1, 1≤i,j,i_{1}...i_{s}≤N} .

Here h = 0 and h = s − 1 correspond to the multi-indeces ji_{2} … i_{s} and i_{1} … i_{s−1}j. By Theorem 3.6 (and Cor. 3.2), and where the target space ℝ^{d} is being replaced by the finite dimensional vector space E_{s}^{n}, for any D ∈ O(Ω)

G(v; D) = ∫_{D} Q_{A}g(x, v(x)) dx,

where for a.e. x ∈ Ω and for all v ∈ E_{s}^{n},

Q_{A}g(x, v) := inf { ∫_{Q} g(x, v + w(y)) dy : w ∈ C_{1-per}^{∞}(ℝ^{N}; E_{s}^{n}) ∩ ker A, ∫_{Q} w(y) dy = 0}·

As shown in [22],

{w ∈ C_{1-per}^{∞}(ℝ^{N}; E_{s}^{n}) : Aw = 0, ∫_{Q} w dx = 0} = {∇^{s}φ : φ ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{n})} ·  (4.1)

Hence

Q_{A}g(x, v) = inf { ∫_{Q} g(x, v + ∇^{s}φ(y)) dy : φ ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{N})}·

In particular

G(∇^{s}u; D) = ∫_{D} Q^{s}f(x, u, …, ∇^{s}u) dx.  (4.2)

Let {u_{k}} ⊂ W^{s,p}(Ω; ℝ^{n}) be any sequence such that u_{k} ⇀ u in W^{s,p}(Ω; ℝ^{n}). Extracting a subsequence, if necessary, we may assume that

u_{k} := (u_{k}, …, ∇^{s−1}u_{k}) → u := (u, …, ∇^{s−1}u) in L^{p}(D; E_{[s−1]}^{n}).

Since ∇^{s}u_{k} ⇀ ∇^{s}u in L^{p}(D; E_{s}^{n}) and A∇^{s}u_{k} = 0, by Lemma 3.1 there exists a p-equi-integrable sequence {V_{k}} ⊂ L^{p}(D; E_{s}^{n}) ∩ ker A such that V_{k} ⇀ ∇^{s}u in L^{p}(D; E_{s}^{n}) and

lim inf_{k→∞} ∫_{D} g(x, V_{k}(x)) dx ≤ lim inf_{k→∞} ∫_{Ω} f(x, u_{k}, …, ∇^{s}u_{k}) dx.

Thus

G(∇^{s}u; D) ≤ F(u; D).  (4.3)

To prove the converse inequality, fix x_{0} ∈ Ω and r > 0, and consider any p-equi-integrable sequence {V_{k}} ⊂ L^{p}(B(x_{0}; r); E_{s}^{n}) ∩ ker A such that V_{k} ⇀ ∇^{s}u in L^{p}(B(x_{0}; r); E_{s}^{n}). An induction argument, similar to the one used in [22] to prove (4.1) above, shows that AV_{k} = 0 if and only if there exists φ_{k} ∈ W^{s,p}(B(x_{0}; r); ℝ^{n}) such that ∇^{s}φ_{k} = V_{k}. By Lemmas 1.1–1.3 in [24], for any φ ∈ W^{s,p}(B(x_{0}; r); ℝ^{n}) we may find a unique function P ∈ C^{∞}(ℝ^{N}; ℝ^{n}) whose components are polynomials of degree s − 1 such that

∫_{B(x_{0},r)} ∇^{l}(φ − P) dx = 0  0 ≤ l ≤ s − 1,  (4.4)

and a constant C(n, N, s, p, r) > 0 such that the following Poincaré type inequality holds

||φ − P ||_{W^{s,p}(B(x_{0};r);ℝ^{n})} ≤ C||∇^{s}φ||_{L^{p}(B(x_{0};r);E_{s}^{n})}.  (4.5)

Let P_{k} and P be the functions associated to φ_{k} and u, respectively, and satisfying (4.4, 4.5). Since ∇^{s}φ_{k} ⇀ ∇^{s}u in L^{p}(B(x_{0}; r); E_{s}^{n}), we have that

φ_{k} − P_{k} ⇀ u − P in W^{s,p}(B(x_{0}; r); ℝ^{n}),

so

u_{k} := φ_{k} − P_{k} + P ⇀ u in W^{s,p}(B(x_{0}; r); ℝ^{N}).

Consider a subsequence of {V_{k}} (not relabelled) such that the two sequences

{(u_{k}, …, ∇^{s−1}u_{k}, V_{k})} and {(u, …, ∇^{s−1}u, V_{k})}

generate the Young measure {δ_{(u(x),... ,∇^{s−1}u)} ⊗ ν_{x}}_{x∈B(x_{0},r)}, and

(u_{k}, …, ∇^{s−1}u_{k}) → (u, …, ∇^{s−1}u)

pointwise and in L^{p}(B(x_{0}; r); E_{s−1}^{n}). Since {V_{k}} is p-equi-integrable and u_{k} converge to u strongly in W^{s−1,p}(Ω; ℝ^{N}), it follows from Theorem 2.1 and the growth condition on f that

lim_{k→∞} ∫_{B(x_{0},r)} f(u_{k}, …, ∇^{s−1}u_{k}, V_{k}) dx = lim_{k→∞} ∫_{B(x_{0},r)} g(x, V_{k}(x)) dx.

Thus

G(∇^{s}u; B(x_{0}, r)) ≥ F(u; B(x_{0}, r)),

which, together with (4.3), yields

G(∇^{s}u; B(x_{0}, r)) = F(u; B(x_{0}, r)).  (4.6)

Since F(u; ·) and G(∇^{s}u; ·) are both traces of a Radon measures absolutely continuous with respect to L^{N}⌊Ω, by (4.2) and (4.6) we immediately obtain that

F(u; D) = G(∇^{s}u; D) = ∫_{D} Qf(x, u, …, ∇^{s}u) dx.

Proof of Theorem 1.5. We only proof Theorem 1.5 for 1 ≤ p < ∞, the case p = ∞ being very similar. For v ∈ ℝ^{N^{2}−1} let

v = (v^{(1)}, …, v^{(N)}), where v^{(i)} ∈ ℝ^{N}, i = 1, …, N − 1, v^{(N)} ∈ ℝ^{N−1}.

Given a function v ∈ L^{p}(Ω; ℝ^{N^{2}−1}) define the differential operator A as follows

Av := ( curl v^{(1)} ⋮ curl v^{(N−1)} curl (v^{(N)}, −v_{1}^{(1)} − … − v_{N−1}^{(N−1)})) .

A straightforward calculation shows that A satisfies the constant-rank property (1.1).

Given a Carathéodory function f : Ω×ℝ^{N} ×ℝ^{N^{2}} → [0, ∞), we define f̂(x, u, v), for (x, u, v) ∈ Ω×ℝ^{N} ×ℝ^{N^{2}−1}, as

f̂(x, u, v) = f (x, u, (v^{(1)}, …, v^{(N−1)}, (\begin{matrix}v^{(N)} \\ −v_{1}^{(1)} − … − v_{N−1}^{(N−1)} \end{matrix}))) .

Let u ∈ W^{1,p}(Ω; ℝ^{N}), with div u = 0, and let {u_{n}} ⊂ W^{1,p}(Ω; ℝ^{N}) be such that div u_{n} = 0 and u_{n} ⇀ u in W^{1,p}(Ω; ℝ^{N}). By Lemma 3.1 there exists a p-equi-integrable sequence {V_{n}} ⊂ L^{p}(Ω; ℝ^{N^{2}−1}) ∩ ker A such that V_{n} ⇀ v in L^{p}(D; ℝ^{N^{2}−1}) and

lim inf_{n→∞} ∫_{Ω} f̂(x, u, V_{n}) dx ≤ lim inf_{n→∞} ∫_{Ω} f̂(x, u_{n}, v_{n}) dx = lim inf_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}) dx,  (4.7)

where

v_{n} := (∇u_{n}^{(1)}, …, ∇u_{n}^{(N−1)}, ( ⋮_{\frac{∂u_{n}^{(N)}}{∂x_{N−1}}}^{\frac{∂u_{n}^{(N)}}{∂x_{1}}} )) , v := (∇u^{(1)}, …, ∇u^{(N−1)}, ( ⋮_{\frac{∂u^{(N)}}{∂x_{N−1}}}^{\frac{∂u^{(N)}}{∂x_{1}}} )) .  (4.8)

Define

G(v; D) := inf{lim inf_{n→∞} ∫_{D} ĝ(x, V_{n}(x)) dx : {V_{n}} ⊂ L^{p}(D; ℝ^{N^{2}−1}) ∩ ker A is p-equi-integrable,
and V_{n} ⇀ v in L^{p}(D; ℝ^{N^{2}−1})},

where ĝ is the Carathéodory function defined by ĝ(x, v) := f̂(x, u(x), v). By Theorem 3.6 (and Cor. 3.2)

G(v; Ω) = ∫_{Ω} Q_{A}ĝ(x, v) dx,  (4.9)

where

Q_{A}ĝ(x, v(x)) := inf { ∫_{Q} f̂(x, u(x), v(x) + w(y)) dy : w ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{N^{2}−1}) ∩ ker A,

∫_{Q} w(y) dy = 0}·

24 A. BRAIDES, I. FONSECA AND G. LEONI

Now

w ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{N^{2}−1}) ∩ ker A and ∫_{Q} w(y) dy = 0

if and only if there exists φ ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{N}) such that

w = (∇φ^{(1)}, …, ∇φ^{(N−1)}, ( ⋮_{\frac{∂φ^{(N)}}{∂x_{N−1}}}^{\frac{∂φ^{(N)}}{∂x_{1}}} ))

and \frac{∂φ^{(N)}}{∂x_{N}} = −\frac{∂φ^{(1)}}{∂x_{1}} − … − \frac{∂φ^{(N−1)}}{∂x_{N−1}} . Hence

Q_{A}ĝ(x, v(x)) = inf { ∫_{Q} f(x, u(x), ∇u(x) + ∇φ(y)) dy : φ ∈ C_{1-per}^{∞}(ℝ^{N}; ℝ^{N}),
div φ = 0} = f̄(x, u(x), ∇u(x)).

(4.10)

Thus, by (4.7, 4.9), and (4.10),

∫_{Ω} f̄(x, u(x), ∇u(x)) dx = G(v; Ω) ≤ lim inf_{n→∞} ∫_{Ω} f̂(x, u, V_{n}) dx ≤ lim inf_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) dx,

and, in turn,

∫_{Ω} f̄(x, u(x), ∇u(x)) dx ≤ inf { lim inf_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) dx : {u_{n}} ⊂ W^{1,p}(Ω; ℝ^{N}),
div u_{n} = 0, u_{n} ⇀ u in W^{1,p}(Ω; ℝ^{N})}·

To prove the converse inequality, fix ε > 0. By the definition of G(v; Ω), there exists a p-equi-integrable sequence {V_{n}} ⊂ L^{p}(D; ℝ^{N^{2}−1}) ∩ ker A such that V_{n} ⇀ v in L^{p}(D; ℝ^{N^{2}−1}) and

∫_{Ω} f̄(x, u(x), ∇u(x)) dx + ε > lim inf_{n→∞} ∫_{D} ĝ(x, V_{n}(x)) dx = lim inf_{n→∞} ∫_{D} f̂(x, u(x), V_{n}(x)) dx,  (4.11)

where we used for v the notation introduced in (4.8). Now AV_{n} = 0 if and only if there exists φ_{n} ∈ W^{1,p}(Ω; ℝ^{N}) such that

V_{n} = (∇φ_{n}^{(1)}, …, ∇φ_{n}^{(N−1)}, ( ⋮_{\frac{∂φ_{n}^{(N)}}{∂x_{N−1}}}^{\frac{∂φ_{n}^{(N)}}{∂x_{1}}} ))

and \frac{∂φ_{n}^{(N)}}{∂x_{N}} = −\frac{∂φ_{n}^{(1)}}{∂x_{1}} − … − \frac{∂φ_{n}^{(N−1)}}{∂x_{N−1}} . Since ∇φ_{n} ⇀ ∇u in L^{p}(Ω; ℝ^{N^{2}}), we have that

φ_{n} − \frac{1}{|Ω|} ∫_{Ω} φ_{n}(x) dx ⇀ U in W^{1,p}(Ω; ℝ^{N}),

where U = u + c for some constant c ∈ ℝ^{N}. So

u_{n} := φ_{n} − \frac{1}{|Ω|} ∫_{Ω} φ_{n}(x) dx − c ⇀ u in W^{1,p}(Ω; ℝ^{N}),

and div u_{n} = 0. Consider a subsequence {V_{n_{k}}} of {V_{n}} such that

lim_{k→∞} ∫_{Ω} f̂(x, u(x), V_{n_{k}}(x)) dx = lim inf_{n→∞} ∫_{Ω} f̂(x, u(x), V_{n}(x)) dx

and {(u_{n_{k}}, V_{n_{k}})} and {(u, V_{n_{k}})} generates the Young measure {δ_{u(x)} ⊗ ν_{x}}_{x∈Ω}. Since {V_{n_{k}}} is p-equi-integrable and u_{n_{k}} converge to u strongly in L^{p}(Ω; ℝ^{N}), it follows from Theorem 2.1 and the growth condition on f that

lim_{k→∞} ∫_{Ω} f̂(x, u(x), V_{n_{k}}(x)) dx = lim_{k→∞} ∫_{Ω} f̂(x, u_{n_{k}}(x), V_{n_{k}}(x)) dx.

By (4.11)

∫_{Ω} f̄(x, u(x), ∇u(x)) dx + ε > lim_{k→∞} ∫_{Ω} f̂(x, u_{n_{k}}(x), V_{n_{k}}(x)) dx = lim_{k→∞} ∫_{Ω} f(x, u_{n_{k}}(x), ∇u_{n_{k}}(x)) dx

≥ inf { lim inf_{n→∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x)) dx : {u_{n}} ⊂ W^{1,p}(Ω; ℝ^{N}),

div u_{n} = 0, u_{n} ⇀ u in W^{1,p}(Ω; ℝ^{N})}·

It now suffices to let ε → 0^{+}.

5. Homogenization

In this section we will limit our analysis to the case where 1 < q < ∞.

Lemma 5.1. Let f : ℝ^{N}×ℝ^{d} → [0, ∞) be a continuous function satisfying (A_{1})-(A_{2}). Let v ∈ L^{q}(D; ℝ^{d})∩ker A, where D ∈ O(Ω), ε_{k} → 0^{+}, and let {v̂_{k}} ⊂ L^{q}(D_{1}; ℝ^{d}) be a sequence of functions such that

v̂_{k} ⇀ v in L^{q}(D_{1}; ℝ^{d}), Av̂_{k} → 0 in W^{−1,q}(D_{1}; ℝ^{l}),

for some D_{1} ∈ O(Ω), with D_{1} ⊂ D. Then we can find a q-equi-integrable sequence {v_{k}} ⊂ L^{q}(D; ℝ^{d}) ∩ ker A such that ∫_{D} v_{k} dx = ∫_{D} v dx,

v_{k} ⇀ v in L^{q}(D; ℝ^{d}),  ||v̂_{k} − v_{k}||_{L^{s}(D_{1})} → 0 for all 1 ≤ s < q  (5.1)

and

lim inf_{k→∞} ∫_{D_{1}} f (x/ε_{k}, v_{k}(x)) dx ≤ lim inf_{k→∞} ∫_{D_{1}} f (x/ε_{k}, v̂_{k}(x)) dx,
lim sup_{k→∞} ∫_{D\D_{1}} |v_{k}(x)|^{q} dx ≤ ∫_{D\D_{1}} |v(x)|^{q} dx.

(5.2)

Moreover, if D = Q, then v_{k} = v + w_{k}, with w_{k} ∈ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A.

Remark 5.2. Lemma 5.1 implies, in particular, that for every v ∈ L^{q}(Ω; ℝ^{d}) ∩ ker A

Γ − lim inf_{n→∞} F_{ε_{n}}(v; D) = inf{lim inf_{n→∞} F_{ε_{n}}(v_{n}; D) : v_{n} ∈ L^{q}(D; ℝ^{d}) ∩ ker A,
v_{n} ⇀ v in L^{q}(D; ℝ^{d}), ∫_{D} v_{n} dx = ∫_{D} v dx},

and if D = Q then

Γ − lim inf_{n→∞} F_{ε_{n}}(v; Q) = inf{lim inf_{n→∞} F_{ε_{n}}(v + w_{n}; Q) : w_{n} ∈ L_{1−per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A
w_{n} ⇀ 0 in L^{q}(Q; ℝ^{d}), ∫_{Q} w_{n} dx = 0}·

Proof of Lemma 5.1. Let g(x) := x in Q and extend it periodically to ℝ^{N} with period 1. Set g_{k}(x) := g(x/ε_{k}). Since {g_{k}} is bounded in L^{∞} and v̂_{k} ⇀ v in L^{q}(D_{1}; ℝ^{d}), by Theorem 2.1 there exists a subsequence {ε_{n}} of {ε_{k}} such that

{(g_{n}(x), v̂_{n}(x))} generates a Young measure {ν_{x}}

and

lim_{n→∞} ∫_{D_{1}} f (x/ε_{n}, v̂_{n}(x)) dx = lim inf_{k→∞} ∫_{D_{1}} f (x/ε_{k}, v̂_{k}(x)) dx.

For i ∈ ℕ let

F_{i} := \{x ∈ D_{1} : dist(x, ∂D_{1}) < \frac{1}{i}\}

and consider cut-off functions θ_{i} with compact support in D_{1} and such that θ_{i} ≡ 1 in D_{1} \ F_{i}. Set w_{i,n} := θ_{i}(v̂_{n} − v) ∈ L^{q}(Q; ℝ^{d}). Then we can proceed as in the proof of Lemma 3.1 to find a q-equi-integrable sequence {v_{i} := v + w_{i}}, where {w_{i}} satisfies (3.3, 5.1) holds, and the two sequences

{(g_{n_{i}}(x), v_{i}(x))}  and  {(g_{n_{i}}(x), v̂_{n_{i}}(x))}

generate the same Young measure {ν_{x}}. Hence by Theorem 2.1

lim_{i→∞} ∫_{D_{1}} f (x/ε_{n_{i}}, v_{i}(x)) dx = ∫_{D_{1}} (∫_{ℝ^{N} ×ℝ^{d}} f(X, V)dν_{x}(X, V)) dx ≤ lim_{i→∞} ∫_{D_{1}} f (x/ε_{n_{i}}, v̂_{n_{i}}(x)) dx

= lim inf_{k→∞} ∫_{D_{1}} f (x/ε_{k}, v̂_{k}(x)) dx,

where we have used (A_{2}), and the facts that {v_{i}(x)} is q-equi-integrable over D_{1}, and that f is a continuous function.

To prove the second inequality in (5.2), we remark that by (3.3) and the fact that ŵ_{i} = θ_{i}(v̂_{n_{i}} − v) ≡ 0 outside D_{1}, we have for all 1 ≤ s < q

||v_{i} − v||_{L^{s}(D\D_{1})} =||ŵ_{i} − w_{i}||_{L^{s}(D\D_{1})} → 0.

Hence {v_{i}(x)} generates the Young measure {μ_{x} = δ_{v(x)}} on D \ D_{1}, and since {v_{i}} is q-equi-integrable we have that

lim sup_{i→∞} ∫_{D\D_{1}} |v_{i}(x)|^{q} dx = ∫_{D\D_{1}} |Y |^{q}dμ_{x}(Y)dx = ∫_{D\D_{1}} |v(x)|^{q} dx.

To complete the proof it suffices to define v_{k} := v_{n_{i}} for each n_{i} ≤ k < n_{i+1}. Clearly

lim inf_{k→∞} ∫_{D_{1}} f (x/ε_{k}, v_{k}(x)) dx ≤ lim inf_{i→∞} ∫_{D_{1}} f (x/ε_{n_{i}}, v_{i}(x)) dx.

Lemma 5.3. Let ε_{n} → 0^{+} and let R(Ω) be the family of all finite unions of open cubes contained in Ω and with vertices in ℚ^{N}. Then there exists a subsequence {ε_{n_{k}}} of {ε_{n}} such that the Γ–limit

Γ − lim_{k→∞} F_{ε_{n_{k}}} (v; R)

exists for all v ∈ L^{q}(R; ℝ^{d}) ∩ ker A and for all R ∈ R(Ω).

Proof of Lemma 5.3. Fix R ∈ R(Ω). For simplicity set F_{n} := F_{ε_{n}} and let B denote the closed unit ball of L^{q}(R; ℝ^{d}). For each l ∈ ℕ consider

l B := {v ∈ L^{q}(R; ℝ^{d}) : ||v||_{L^{q}} ≤ l}·

Since q > 1 the dual of L^{q}(R; ℝ^{d}) is separable, and hence the space l B endowed with the weak topology is metrizable. Let d_{l} be any metric which generates the L^{q}-weak topology. Consider l = 1 and apply Proposition 2.4 to the sequence of functionals {F_{n}(·; R)} restricted to (B ∩ ker A, d_{1}). Then we can find an increasing sequence of integers {n_{j}^{1}} such that

Γ(d_{1}) − lim_{j→∞} F_{n_{j}^{1}}(v; R)

exists for all v ∈ B ∩ ker A. We now proceed recursively, so that given l ∈ ℕ we apply Proposition 2.4 to the sequence of functionals {F_{n_{j}^{l−1}}(·; R)} restricted to (l B ∩ ker A, d_{l}) to obtain a subsequence {n_{j}^{l}} of {n_{j}^{l−1}} such that

Γ(d_{l}) − lim_{j→∞} F_{n_{j}^{l}}(v; R)

exists for all v ∈ l B ∩ ker A. Let n_{k} := n_{k}^{k}. Since {n_{k}} is a subsequence of all {n_{j}^{l}} we have that for each l ∈ ℕ

Γ(d_{l}) − lim_{k→∞} F_{n_{k}}(v; R)

exists for all v ∈ l B ∩ ker A.

We claim that the Γ–limit

Γ − lim_{k→∞} F_{n_{k}}(v; R)  (5.3)

exists for all v ∈ L^{q}(R; ℝ^{d}) ∩ ker A. Indeed assume by contradiction that this is not the case. Then there exists v ∈ L^{q}(R; ℝ^{d}) ∩ ker A for which

F^{−}(v; R) := Γ − lim inf_{k→∞} F_{n_{k}}(v; R) < F^{+}(v; R) := Γ − lim sup_{k→∞} F_{n_{k}}(v; R).

Let v_{k} ∈ L^{q}(R; ℝ^{d}) ∩ ker A be such that v_{k} ⇀ v in L^{q}(R; ℝ^{d}) and

lim inf_{k→∞} F_{n_{k}}(v_{k}; R) = F^{−}(v; R).

Since v_{k} ⇀ v in L^{q}(R; ℝ^{d}), we may find an integer l_{0} such that v_{k}, v ∈ l_{0} B ∩ ker A for all k ∈ ℕ. Consequently

d_{l_{0}}(v_{k}, v) → 0 as k → ∞,

and thus

Γ(d_{l_{0}}) − lim inf_{k→∞} F_{n_{k}}(v; R) ≤ lim inf_{k→∞} F_{n_{k}}(v_{k}; R) = F^{−}(v; R) < F^{+}(v; R) ≤ Γ(d_{l_{0}}) − lim sup_{k→∞} F_{n_{k}}(v; R),

which contradicts the existence of the Γ-limit Γ(d_{l_{0}}) − lim_{k→∞} F_{n_{k}}(v; R), and where we have used the fact that

F^{+}(v; R) = inf {lim sup_{k→∞} F_{n_{k}}(z_{k}; R) : z_{k} ∈ L^{q}(R; ℝ^{d}) ∩ ker A, z_{k} ⇀ v in L^{q}(R; ℝ^{d})}

≤ Γ(d_{l_{0}}) − lim sup_{k→∞} F_{n_{k}}(v; R)

= inf {lim sup_{k→∞} F_{n_{k}}(z_{k}; R) : z_{k} ∈ l_{0} B ∩ ker A, z_{k} ⇀ v in L^{q}(R; ℝ^{d})} ·

Hence (5.3) holds. To conclude the proof of the lemma it suffices to observe that since the family R(Ω) is countable, with a diagonal process it is possible to extract a further subsequence for which (5.3) holds for all R ∈ R(Ω).

Remark 5.4. The previous proof asserts that for any given D ∈ O(Ω) and ε_{n} → 0^{+} there exists a subsequence {ε_{n_{k}}} (depending on the particular set D) of {ε_{n}} such that such that the Γ–limit

Γ − lim_{k→∞} F_{ε_{n_{k}}} (v; D)

exists for all v ∈ L^{q}(D; ℝ^{d}) ∩ ker A.

Lemma 5.5. Assume that conditions (A_{1})-(A_{2}) hold. Given ε_{n} → 0^{+}, let {ε_{n_{k}}} be as in Lemma 5.3, and for any D ∈ O(Ω) set

F^{−}(·; D) := Γ − lim inf_{k→∞} F_{ε_{n_{k}}} (·; D).

Then F^{−}(v; ·) is the trace of a Radon measure.

Proof of Lemma 5.5. We start by establishing inner regularity. Precisely, we claim that for any v ∈ L^{q}(D; ℝ^{d})∩ ker A and D ∈ O(Ω)

F^{−}(v; D) = sup {F^{−}(v; R) : R ∈ R(Ω), R ⊂ D} = lim_{R↗D} F^{−}(v; R),  (5.4)

where the limit is taken over all finite unions of cubes R ∈ R(Ω) with R ⊂ D. For fixed η > 0 there exists δ > 0 such that

∫_{D_{0}} C(1 + |v(x)|^{q}) dx < η  (5.5)

for any measurable set D_{0} ⊂ D, with |D_{0}| < δ, and where C is the constant given in (A_{2}). Let R ∈ R(Ω), with R ⊂ D and |D \ R| < δ, and, in light of Lemma 5.3, consider a sequence {v̂_{k}} ⊂ L^{q}(R; ℝ^{d}) ∩ ker A, with v̂_{k} ⇀ v in L^{q}(R; ℝ^{d}), and such that

lim_{k→∞} F_{ε_{n_{k}}} (v̂_{k}; R) = F^{−}(v; R).

By Lemma 5.1 there exists a q-equi-integrable sequence {v_{k}} ⊂ L^{q}(D; ℝ^{d}) ∩ ker A such that

v_{k} ⇀ v in L^{q}(D; ℝ^{d}), ∫_{D} v_{k} dx = ∫_{D} v dx,

and

lim inf_{k→∞} ∫_{R} f (x/ε_{n_{k}}, v_{k}(x)) dx ≤ lim_{k→∞} ∫_{R} f (x/ε_{n_{k}}, v̂_{k}(x)) dx,
lim sup_{k→∞} ∫_{D\R} |v_{k}(x)|^{q} dx ≤ ∫_{D\R} |v(x)|^{q} dx.

Hence

F^{−}(v; D) ≤ lim inf_{k→∞} ∫_{D} f (x/ε_{n_{k}}, v_{k}(x)) dx ≤ lim_{k→∞} ∫_{R} f (x/ε_{n_{k}}, v̂_{k}(x)) dx + lim sup_{k→∞} ∫_{D\R} C(1 + |v_{k}(x)|^{q}) dx
≤ F^{−}(v; R) + ∫_{D\R} C(1 + |v(x)|^{q}) dx ≤ F^{−}(v; R) + η,

where we have used (A_{2}) and (5.5). Consequently

F^{−}(v; D) ≤ sup {F^{−}(v; R) : R ∈ R(Ω), R ⊂ D} + η,

and letting η → 0^{+} we obtain one inequality in (5.4). To show the opposite inequality, note that if {v_{k}} ⊂ L^{q}(D; ℝ^{d}) ∩ ker A, with v_{k} ⇀ v in L^{q}(D; ℝ^{d}), then the restriction of v_{k} to R belongs to L^{q}(R; ℝ^{d}) ∩ ker A, and v_{k} ⇀ v in L^{q}(R; ℝ^{d}). Therefore

F^{−}(v; R) ≤ lim inf_{k→∞} ∫_{R} f (x/ε_{n_{k}}, v_{k}(x)) dx ≤ lim inf_{k→∞} ∫_{D} f (x/ε_{n_{k}}, v_{k}(x)) dx

and by taking the infimum over all such sequences we get that

F^{−}(v; R) ≤ F^{−}(v; D),  (5.6)

and in turn (5.4) holds.

In order to prove that F^{−}(v; ·) is the trace of a Radon measure, as it is usual it suffices to prove subadditivity for nested sets (see [3, 21]). Let B ⊂⊂ C ⊂⊂ D. By (5.4) for fixed η > 0 we find R ∈ R(Ω) such that R ⊂ D and

F^{−}(v; D) ≤ η + F^{−}(v; R).

Construct R_{1}, R_{2} ∈ R(Ω) with

R ⊂ R_{1} ∪ R_{2}, R_{1} ⊂ D \ \overline{B} and R_{2} ⊂ C.

By (5.6) we have

F^{−}(v; D) ≤ η + F^{−}(v; R) ≤ η + F^{−}(v; R_{1} ∪ R_{2}).  (5.7)

By the definition of Γ-convergence and Lemma 5.1 there exist v_{k} ∈ L^{q}(R_{1}; ℝ^{d}) ∩ ker A and w_{k} ∈ L^{q}(R_{2}; ℝ^{d}) ∩ ker A, with v_{k} ⇀ v in L^{q}(R_{1}; ℝ^{d}) and w_{k} ⇀ v in L^{q}(R_{2}; ℝ^{d}), such that

F^{−}(v; R_{1}) = lim_{k→∞} Fε̃_{n_{k}} (v_{k}; R_{1}),  F^{−}(v; R_{2}) = lim_{k→∞} Fε̃_{n_{k}} (w_{k}; R_{2}),  (5.8)

where {ε̃_{n_{k}}} is a subsequence of {ε_{n_{k}}} and {v_{k}}, {w_{k}} are q-equi-integrable over R_{1} and R_{2}, respectively. Let θ_{j} be smooth cut-off functions which are equal to 1 on B and 0 on D \ \overline{C}, and such that |\{0 < θ_{j} < 1\}| → 0 as j → ∞. Set

V̂_{j,k} := (1 − θ_{j})v_{k} + θ_{j}w_{k}.

For j fixed

AV̂_{j,k} = (1 − θ_{j})Av_{k} + θ_{j}Aw_{k} − ∑_{i=1}^{N} A^{(i)}v_{k} \frac{∂θ_{j}}{∂x_{i}} + ∑_{i=1}^{N} A^{(i)}w_{k} \frac{∂θ_{j}}{∂x_{i}} → 0

in W^{−1,q}(R_{1} ∪ R_{2}; ℝ^{l}) strong, because ||v_{k} − w_{k}||_{W^{−1,q}(B\\overline{C};ℝ^{l})} → 0 as k → ∞. Diagonalize to get V̂_{j} := V̂_{j,k_{j}} such that

V̂_{j} ⇀ v in L^{q}(R_{1} ∪ R_{2}; ℝ^{d}), AV̂_{j} → 0 in W^{−1,q}(R_{1} ∪ R_{2}; ℝ^{l}).

By Lemma 5.1 we can find V_{j} ∈ L^{q}(R_{1} ∪ R_{2}; ℝ^{d}) ∩ ker A such that V_{j} ⇀ v in L^{q}(R_{1} ∪ R_{2}; ℝ^{d}) and

lim inf_{j→∞} ∫_{R_{1}∪R_{2}} f (x/ε̃_{n_{k_{j}}} , V_{j}(x)) dx ≤ lim inf_{j→∞} ∫_{R_{1}∪R_{2}} f (x/ε̃_{n_{k_{j}}} , V̂_{j}(x)) dx.

Consequently, by (5.7)

F^{−}(v; D) ≤ η + F^{−}(R_{1} ∪ R_{2}; D) ≤ η + lim inf_{j→∞} ∫_{R_{1}∪R_{2}} f (x/ε̃_{n_{k_{j}}} , V_{j}(x)) dx

≤ η + lim inf_{j→∞} ∫_{R_{1}∪R_{2}} f (x/ε̃_{n_{k_{j}}} , V̂_{j}(x)) dx ≤ η + lim sup_{j→∞} ∫_{R_{1}} f (x/ε̃_{n_{k_{j}}} , v_{k_{j}}(x)) dx

+ lim sup_{j→∞} ∫_{R_{2}} f (x/ε̃_{n_{k_{j}}} , w_{k_{j}}(x)) dx + lim sup_{j→∞} ∫_{{0<θ_{j}<1}} C(1 + |w_{k_{j}}(x)|^{q} + |v_{k_{j}}(x)|^{q})

≤ η + F^{−}(v; R_{1}) + F^{−}(v; R_{2}) ≤ η + F^{−}(v; D \ \overline{B}) + F^{−}(v; C),

where we have used (5.6) and the fact that in (5.8) inferior limits are actually limits. It now suffices to let η → 0^{+}.

Lemma 5.6. Under conditions (A_{1})-(A_{2}), for L^{N} a.e. x_{0} ∈ Ω we have

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) = f_{hom}(v(x_{0})).

Proof of Lemma 5.6. We divide the proof in three steps.

Given ε_{n} → 0^{+}, let {ε_{n_{k}}} be as in Lemma 5.3. In order to simplify the notations, in the proof of this lemma we will represent {ε_{n_{k}}} simply by {ε}.

Step 1. We claim that

F^{−}(v(· − x_{0}); D + x_{0}) = F^{−}(v; D).

The proof is similar to the one of Lemma 3.9 in [11]. We present it here for the convenience of the reader. Let v_{ε} ∈ L^{q}(D; ℝ^{d}) ∩ ker A be such that v_{ε} ⇀ v in L^{q}(D; ℝ^{d}) and

F^{−}(v; D) = lim inf_{ε→0} F_{ε}(v_{ε}; D).  (5.9)

Consider the sequence z_{ε} := [x_{0}/ε] ∈ ℤ^{N}, so that x_{ε} := z_{ε} ε converges to x_{0}. Here [z] := ([z_{1}], … [z_{N}]), with [z_{i}] denoting the integer part of z_{i} ∈ ℝ. By the periodicity of f,

F_{ε}(v_{ε}; D) = ∫_{D} f (\frac{x + x_{ε}}{ε} , v_{ε}(x)) dx = ∫_{D+x_{ε}} f (\frac{y}{ε} , v_{ε}(y − x_{ε})) dy.

Let B ⊂⊂ D. For ε sufficiently small we have that D + x_{ε} ⊃ B + x_{0}, and thus

F_{ε}(v_{ε}; D) ≥ ∫_{B+x_{0}} f (\frac{y}{ε} , v_{ε}(y − x_{ε})) dy.  (5.10)

Since v_{ε}(· − x_{ε}) ⇀ v(· − x_{0}) in L^{q}(B + x_{0}; ℝ^{d}), and v_{ε}(· − x_{ε}) ∈ L^{q}(B + x_{0}; ℝ^{d}) ∩ ker A, by (5.9, 5.10), we obtain

F^{−}(v; D) ≥ F^{−}(v(· − x_{0}); B + x_{0}).

By letting R ↗ D + x_{0}, R ∈ R(Ω), setting B := R − x_{0} above, we obtain by (5.4)

F^{−}(v; D) ≥ F^{−}(v(· − x_{0}); D + x_{0}).

The converse inequality follows in a similar way. Step 2. Next, we show that

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) ≥ f_{hom}(v(x_{0})) for L^{N} a.e. x_{0} ∈ Ω.

Fix x_{0} ∈ Ω such that

lim_{r→0^{+}} \frac{1}{r^{N}} ∫_{Q(x_{0},r)} |v(x) − v(x_{0})|^{q} dx = 0  (5.11)

and

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) = lim_{r→0^{+}} lim inf_{ε→0^{+}} \frac{1}{r^{N}} ∫_{rQ} f (\frac{x}{ε}, v_{ε,r}(x)) dx < ∞,

where we have used Step 1 and Lemma 5.5, and where we have chosen the radii r → 0^{+} such that F^{−}(v(· + x_{0}); ∂(rQ)) = 0. Here v_{ε,r} ∈ L^{q}(rQ; ℝ^{d}) ∩ ker A and v_{ε,r} ⇀ v(· + x_{0}) in L^{q}(rQ; ℝ^{d}) as ε → 0^{+}. Then

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) = lim_{r→0^{+}} lim inf_{ε→0^{+}} ∫_{Q} f (\frac{r}{ε} y, v(x_{0}) + w_{ε,r}(y)) dy

where w_{ε,r}(y) := v_{ε,r}(ry) − v(x_{0}). As in the proof of Lemma 3.5, we have that w_{ε,r} ⇀ 0 in L^{q}(Q; ℝ^{d}) if we first let ε → 0 and then r → 0^{+}. Diagonalize to get ŵ_{k} ∈ L^{q}(Q; ℝ^{d}) ∩ ker A such that ŵ_{k} ⇀ 0 in L^{q}(Q; ℝ^{d}),

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) = lim_{k→∞} ∫_{Q} f(s_{k} y, v(x_{0}) + ŵ_{k}(y)) dy,

and where s_{k} := 1/ε_{k} → ∞. By Lemma 5.1, applied to the Carathéodory function h(x, v) := f(x, v(x_{0}) + v), there exists a q-equi-integrable sequence {w_{k}} ⊂ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A such that

w_{k} ⇀ 0 in L^{q}(Q; ℝ^{d}), ∫_{Q} w_{k} dy = 0,

and

lim inf_{k→∞} ∫_{Q} f(s_{k} y, v(x_{0}) + w_{k}(y)) dy ≤ lim_{k→∞} ∫_{Q} f(s_{k} y, v(x_{0}) + ŵ_{k}(y)) dy.

Consequently

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) ≥ lim inf_{k→∞} ∫_{Q} f(s_{k} y, v(x_{0}) + w_{k}(y)) dy ≥ lim inf_{i→∞} lim inf_{k→∞} ∫_{Q} f(s_{k} y, v(x_{0}) + θ_{i}(y)w_{k}(y)) dy,

where 0 ≤ θ_{i} ≤ 1 are smooth cut-off functions with compact support in Q such that θ_{i} ≡ 1 in (1 − 1/i)Q, and where we used the q-equi-integrability of {w_{k}} and (A_{2}). Then θ_{i} w_{k} ⇀ 0 in L^{q}(Q; ℝ^{d}) as k → ∞ and i → ∞, in this order, and

lim_{i→∞} lim_{k→∞} A(θ_{i} w_{k}) = 0 in W^{−1,q}(Q; ℝ^{l}).

Diagonalize to get U_{i} := θ_{i} w_{k_{i}} extended by zero outside Q, such that U_{i} ⇀ 0 in L^{q}(Q; ℝ^{d}), AU_{i} → 0 in W^{−1,q}(Q; ℝ^{l}) as i → ∞, and

lim inf_{i→∞} lim inf_{k→∞} ∫_{Q} f(s_{k} y, v(x_{0}) + θ_{i}(y)w_{k}(y)) dy = lim inf_{i→∞} ∫_{Q} f(s_{k_{i}}, v(x_{0}) + U_{i}(y)) dy.

Thus

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) ≥ lim inf_{i→∞} \frac{1}{s_{k_{i}}^{N}} ∫_{s_{k_{i}} Q} f (x, v(x_{0}) + U_{i} (\frac{x}{s_{k_{i}}} )) dx

≥ lim inf_{i→∞} \frac{1}{s_{k_{i}}^{N}} ∫_{([s_{k_{i}}]+1)Q} f (x, v(x_{0}) + U_{i} (\frac{x}{s_{k_{i}}} )) dx

− lim sup_{i→∞} \frac{1}{s_{k_{i}}^{N}} ∫_{([s_{k_{i}}]+1)Q\s_{k_{i}} Q} f (x, v(x_{0}) + U_{i} (\frac{x}{s_{k_{i}}} )) dx,

where [s_{k_{i}}] denotes the integer part of s_{k_{i}}. We claim that the last limit is zero. Indeed

\frac{1}{s_{k_{i}}^{N}} ∫_{([s_{k_{i}}]+1)Q\s_{k_{i}} Q} f (x, v(x_{0}) + U_{i} (\frac{x}{s_{k_{i}}} )) dx = ∫_{\frac{([s_{k_{i}}]+1)}{s_{k_{i}}} Q\Q} f(s_{k_{i}} y, v(x_{0}) + U_{i}(y)) dy.

Since ([s_{k_{i}}] + 1)/s_{k_{i}} → 1, we have that

|\frac{([s_{k_{i}}] + 1)}{s_{k_{i}}}Q \ Q| = (\frac{([s_{k_{i}}] + 1)}{s_{k_{i}}})^{N} − 1 → 0,

and thus the claim follows from the q-equi-integrability of {U_{i}} and (A_{2}). Hence, setting

m_{i} := 1/s_{k_{i}}, n_{i} := [s_{k_{i}}] + 1 ∈ ℕ,

we obtain

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) ≥ lim inf_{i→∞} \frac{1}{n_{i}^{N}} ∫_{n_{i}Q} f(x, v(x_{0}) + U_{i}(m_{i} x)) dx = lim inf_{i→∞} ∫_{Q} f(n_{i} y, v(x_{0}) + U_{i}(n_{i} m_{i}y)) dy.

We claim that

U_{i}(n_{i} m_{i} ·) ⇀ 0 in L^{q}(Q; ℝ^{d}),  AU_{i}(n_{i} m_{i} ·) → 0 in W^{−1,q}(Q; ℝ^{l})  (5.12)

as i → ∞. Assuming that the claim holds, by Lemma 5.1 there exists a q-equi-integrable sequence {V_{i}} ⊂ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A such that

V_{i} ⇀ 0 in L^{q}(Q; ℝ^{d}), ∫_{Q} V_{i} dy = 0,

and

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) ≥ lim inf_{i→∞} ∫_{Q} f (n_{i} y, v(x_{0}) + U_{i}(n_{i} m_{i}y)) dy ≥ lim inf_{i→∞} ∫_{Q} f (n_{i} y, v(x_{0}) + V_{i}(y)) dy

= lim inf_{i→∞} \frac{1}{n_{i}^{N}} ∫_{n_{i}Q} f (x, v(x_{0}) + V_{i} (\frac{x}{n_{i}^{N}} )) dx ≥ f_{hom}(v(x_{0})),

and where we have used the facts that

V_{i} (\frac{1}{n_{i}^{N}} ·) ∈ L_{n_{i}-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A, ∫_{n_{i}Q} V_{i} (\frac{1}{n_{i}^{N}} y) dy = 0.

Thus it remains to show (5.12). If φ ∈ C_{c}^{∞}(Q) then

\begin{aligned}∫_{Q} U_{i}(n_{i} m_{i} y) φ(y) dy & = \frac{1}{(n_{i} m_{i})^{N}} ∫_{n_{i} m_{i} Q} U_{i}(x) φ (\frac{x}{n_{i} m_{i}} ) dx \\ & = \frac{1}{(n_{i} m_{i})^{N}} (∫_{Q} U_{i}(x) φ(x) dx + ∫_{Q} U_{i}(x) (φ (\frac{x}{n_{i} m_{i}} ) − φ(x)) dx), \end{aligned}

where we have used the fact that U_{i}(x) ≡ 0 in n_{i} m_{i} Q \ Q. Since U_{i} ⇀ 0 in L^{q}(Q; ℝ^{d}) and n_{i} m_{i} → 1 the first integral on the right hand side of the previous inequality tends to zero as i → ∞. By Hölder’s inequality

|∫_{Q} U_{i}(x) (φ (\frac{x}{n_{i} m_{i}} ) − φ(x)) dx| ≤ (sup_{l} ||U_{l}||_{L^{q}(Q)}) (∫_{Q} |φ (\frac{x}{n_{i} m_{i}} ) − φ(x)|^{q′} dx)^{1/q′} .

Since φ is bounded we can apply Lebesgue Dominated Convergence Theorem to conclude that the right hand side approaches zero as i → ∞. In a similar way we can show that

AU_{i}(n_{i} m_{i}·) → 0 in W^{−1,q}(Q; ℝ^{l}) as i → ∞.

We omit the details.

Step 3. To conclude the proof of the lemma it remains to show that

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) ≤ f_{hom}(v(x_{0})) for L^{N} a.e. x_{0} ∈ Ω.

By Remark 1.8(i), for any fixed η > 0 we may find k ∈ ℕ, w ∈ L_{k-per}^{∞}(ℝ^{N}; ℝ^{d}) ∩ ker A such that ∫_{kQ} w dx = 0 and

\frac{1}{k^{N}} ∫_{kQ} f(x, v(x_{0}) + w(x)) dx ≤ f_{hom}(v(x_{0})) + η.  (5.13)

34 A. BRAIDES, I. FONSECA AND G. LEONI

For any fixed r > 0 and for any n ∈ ℕ, let u_{n,r}(x) := w(xnk/r). Then u_{n,r} ∈ L^{∞}(rQ; ℝ^{d}) ∩ ker A, u_{n,r} ⇀ 0 in L^{q}(rQ; ℝ^{d}) as n → ∞, and by Step 1

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) = lim_{r→0^{+}} \frac{F^{−}(v(· + x_{0}); rQ)}{r^{N}} ≤ lim inf_{r→0^{+}} lim inf_{n→∞} \frac{1}{r^{N}} ∫_{rQ} f (\frac{x}{ε_{n,r}} , v(x_{0} + x) + u_{ε}(x)) dx

= lim inf_{r→0^{+}} lim inf_{n→∞} ∫_{Q} f (nk y, v(x_{0} + r y) + w (nk y)) dy,

where ε_{n,r} := r/nk. Since f(·, v) is Q-periodic, there exists δ > 0 such that if |v − v(x_{0})| < δ then

sup_{s∈ℝ^{N}} |f(s, v + w(s)) − f(s, v(x_{0}) + w(s))| < η.

Setting E_{r,δ} := {y ∈ Q : |v(x_{0} + ry) − v(x_{0})| ≥ δ}, we deduce that

\frac{dF^{−}(v; ·)}{dL^{N}}(x_{0}) ≤ η + lim sup_{r→0^{+}} lim sup_{n→∞} ∫_{Q} f (nk y, v(x_{0}) + w (nk y)) dy

+ lim sup_{r→0^{+}} lim sup_{n→∞} ∫_{E_{r,δ}} C (1 + |v(x_{0} + r y)|^{q} + |w (nk y)|^{q}) dy

= η + \frac{1}{k^{N}} ∫_{kQ} f(y, v(x_{0}) + w(y)) dy ≤ f_{hom}(v(x_{0})) + 2η,

where we have used (5.13), the kQ-periodicity of the function h(y) := f(y, v(x_{0}) + w(y)), the equi-integrability of {|u_{ε}|^{q}}, and the fact that (5.11) entails

lim_{r→0^{+}} |{y ∈ Q : |v(x_{0} + r y) − v(x_{0})| ≥ δ})| = 0.

It suffices to let η → 0^{+}.

Proof of Theorem 1.7. We claim that for any ε_{n} → 0^{+}

F_{hom}(·; D) = Γ − lim inf_{n→∞} F_{ε_{n}}(·; D).

By Lemmas 5.5 and 5.6 we always have

F_{hom}(·; D) ≥ Γ − lim inf_{k→∞} F_{ε_{n_{k}}} ≥ Γ − lim inf_{n→∞} F_{ε_{n}}(·; D).

Thus assume for contradiction that that there exists ε_{n} → 0^{+} and v ∈ L^{q}(D; ℝ^{d}) ∩ ker A such that

F_{hom}(v; D) > Γ − lim inf_{n→∞} F_{ε_{n}}(v; D).

Let {v_{n}} ⊂ L^{q}(D; ℝ^{d}) ∩ ker A be such that v_{n} ⇀ v in L^{q}(D; ℝ^{d}) and

F_{hom}(v; D) > lim inf_{n→∞} F_{ε_{n}}(v_{n}; D),

and choose a subsequence {ε_{n_{k}}} such that

F_{hom}(v; D) > lim_{k→∞} F_{ε_{n_{k}}} (v_{n_{k}}; D).

Then, by the previous lemmas, we can extract a further subsequence {ε_{n_{k_{j}}} } such that

F_{hom}(v; D) = Γ − lim inf_{j→∞} F_{ε_{n_{k_{j}}}} (v; D) ≤ lim_{j→∞} F_{ε_{n_{k_{j}}}} (v_{n_{k_{j}}}; D) < F_{hom}(v; D),

which is a contradiction and proves the claim.

Hence it remains to show that, when (A_{3}) holds, for any ε_{n} → 0^{+} and v ∈ L^{q}(D; ℝ^{d}) ∩ ker A

Γ − lim sup_{n→∞} F_{ε_{n}}(v; D) ≤ F_{hom}(v; D).

By taking w_{n} ≡ v and using (A_{2}) we get

Γ − lim sup_{n→∞} F_{ε_{n}}(v; D) ≤ lim sup_{n→∞} F_{ε_{n}}(w_{n}; D) ≤ C ∫_{D}(|v(x)|^{q} + 1)dx.

Hence for any sequence {v_{n}} ⊂ L^{q}(D; ℝ^{d}) ∩ ker A such that v_{n} ⇀ v in L^{q}(D; ℝ^{d}) and

lim sup_{n→∞} F_{ε_{n}}(v_{n}; D) ≤ C ∫_{D}(|v(x)|^{q} + 1)dx,

by (A_{3}) we get that sup_{n} ||v_{n}||_{L^{q}(D)} ≤ L < ∞, where the constant L depends only on the constants in (A_{2}), (A_{3}), and on ||v||_{L^{q}(Ω;ℝ^{d})}. Using the notation introduced in the proof of Lemma 5.3, we conclude that

Γ − lim sup_{n→∞} F_{ε_{n}}(v; D) = Γ(d_{L}) − lim sup_{n→∞} F_{ε_{n}}(v; D).  (5.14)

By Remark 5.4 and by Lemma 5.6, for any subsequence {ε_{n_{k}}} of {ε_{n}} there exists a subsequence {ε_{n_{k_{j}}} } (depending on D) such that

Γ − lim_{j→∞} F_{ε_{n_{k_{j}}}} (v; D) = F_{hom}(v; D).

By (5.14) this implies that

Γ(d_{L}) − lim_{j→∞} F_{ε_{n_{k_{j}}}} (v; D) = F_{hom}(v; D).

We can now apply the second part of Proposition 2.4 in the metric space (L B, d_{L}) to conclude that

Γ(d_{L}) − lim_{n→∞} F_{ε_{n}}(v; D) = F_{hom}(v; D).

Corollary 5.7. Under hypotheses (A_{1})-(A_{3}) the function f_{hom} is A-quasiconvex and the following asymptotic formula holds

f_{hom}(ξ) = lim_{T →+∞} \frac{1}{T^{N}} inf\{∫_{T Q} f(x, ξ + v(x)) dx : v ∈ L_{T -per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A, ∫_{T Q} v(x) dx = 0\} · (5.15)

Proof of Corollary 5.7. It may be shown easily, via a diagonalization procedure and in view of the coercivity condition (A_{3}), that F_{hom}(·; Q) is L^{q}-sequentially weakly lower semicontinuous in ker A. In particular, this entails A-quasiconvexity for f_{hom}. Indeed, fix v ∈ ℝ^{d} and w ∈ L_{1-per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A, with ∫_{Q} w(y) dy = 0, and define w_{n}(x) := w(nx). Then w_{n} ∈ L^{q}(Q; ℝ^{d}) ∩ ker A, w_{n} ⇀ 0 in L^{q}(Q; ℝ^{d}), and so

f_{hom}(v) = F_{hom}(v; Q) ≤ lim inf_{n→∞} F_{hom}(v + w_{n}; Q) = lim inf_{n→∞} ∫_{Q} f_{hom}(v + w(nx)) dx = ∫_{Q} f_{hom}(v + w(x)) dx.

Finally, using Theorem 1.7 (A_{3}), and recalling Remark 5.2, we conclude that

f_{hom}(v) = min{F_{hom}(v + w; Q) : w ∈ L_{1−per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A, ∫_{Q} w(x) dx = 0}

= lim_{ε→0} inf {F_{ε}(v + w; Q) : w ∈ L_{1−per}^{q}(ℝ^{N}; ℝ^{d}) ∩ ker A, ∫_{Q} w(x) dx = 0} ,

and (5.15) follows by setting T = 1/ε and changing variables in the last expression.

As a corollary of Theorem 1.7, we obtain the following result via the same choice of the underlying operator A as in the proof of Theorem 1.5.

Theorem 5.8 (Homogenization with constraint on the divergence). Assume that conditions (A_{1}) − (A_{3}) hold, with d = N^{2}, and let F_{ε} be defined by

F_{ε}(u; D) := ∫_{D} f(\frac{x}{ε}, ∇u) dx

on functions u ∈ W^{1,q}(Ω; ℝ^{N}) such that div u = 0. Then the Γ-limit

F (u; D) := Γ(L^{q}) − lim_{ε→0} F_{ε}(u; D) = ∫_{D} f_{hom}(∇u)dx

exists on functions u ∈ W^{1,q}(Ω; ℝ^{N}) such that div u = 0, where

f_{hom}(v) = inf_{k∈ℕ} \frac{1}{k^{N}} inf\{∫_{kQ} f(x, v + ∇w(x)) dx : w ∈ W_{k-per}^{1,q}(ℝ^{N}; ℝ^{N}), div w = 0\}

for all v ∈ ℝ^{N^{2}}.

Appendix

We prove that in Remark 1.2(iii) in the Introduction

F(v; (a, b)) = F((v_{1}, v_{2}); (a, b)) = ∫_{a}^{b}(ψ^{∗∗}(v_{1}) + v_{2}^{2}) dx,  (5.16)

where ψ^{∗∗}(v_{1}) is the convex envelope of

ψ(v_{1}) := min \{(v_{1} − 1)^{2}, (v_{1} + 1)^{2}\} = \begin{cases}(v_{1} + 1)^{2} & if  v_{1} ≥ 0, \\ (v_{1} − 1)^{2} & if  v_{1} < 0. \end{cases}

Indeed, if v_{1}^{n} ⇀ v_{1} in L^{2}(a, b), v_{2}^{n} ⇀ v_{2} in L^{2}(a, b) and (v_{2}^{n})′ → 0 in H^{−1}(a, b) then the function v_{2} is constant and Jensen’s inequality yields

lim inf_{n→∞} ∫_{a}^{b} f(v_{1}^{n}, v_{2}^{n}) dx ≥ lim inf_{n→∞} ∫_{a}^{b}[ψ(v_{1}^{n}) + (v_{2}^{n})^{2}] dx ≥ lim inf_{n→∞} ∫_{a}^{b} ψ(v_{1}^{n}) dx + lim inf_{n→∞} ∫_{a}^{b}(v_{2}^{n})^{2} dx
≥ ∫_{a}^{b} ψ^{∗∗}(v_{1}) dx + (v_{2})^{2}(b − a).

The arbitrariness of the sequence {(v_{1}^{n}, v_{2}^{n})} allows us to conclude that

F(v; (a, b)) ≥ ∫_{a}^{b} ψ^{∗∗}(v_{1}) dx + (v_{2})^{2}(b − a).

Conversely, suppose that if v_{1} is smooth, v_{1} ∈ L^{2}(a, b), and |{x ∈ (a, b) : v_{1}(x) = 0}| = 0, v_{2} ∈ ℝ, then

F(v; (a, b)) ≤ ∫_{a}^{b} ψ(v_{1}) dx + (v_{2})^{2}(b − a).  (5.17)

Then this inequality remains true for v_{1} ∈ L^{2}(a, b), v_{2} ∈ ℝ arbitrary, because we may approximate v_{1} in L^{2} strong by a sequence {v_{1}^{n}} ⊂ L^{2}(a, b) ∩ C^{∞}(a, b), |{x ∈ (a, b) : v_{1}^{n}(x) = 0}| = 0, and

v_{1} ↦ F((v_{1}, v_{2}); (a, b)), v_{1} ↦ ∫_{a}^{b} ψ(v_{1}) dx

are, respectively, L^{2}-weak lower semicontinuous and L^{2}-strong continuous. Once we establish (5.17) for (v_{1}, v_{2}) ∈ L^{2}(a, b) × ℝ then (5.16) follows because, once again, F((·, v_{2}); (a, b)) is L^{2}-weak lower semicontinuous.

Fix now v_{1} ∈ L^{2}(a, b) ∩ C^{∞}(a, b), with |{x ∈ (a, b) : v_{1}(x) = 0}| = 0, and let v_{2} ∈ ℝ, δ > 0 be fixed. Set

\begin{aligned}A_{δ}^{−} & := \{x ∈ (a, b) : (v_{1}(x) − 1)^{2} < (v_{1}(x) + 1)^{2} − δ\}, \\ A_{δ}^{+} & := \{x ∈ (a, b) : (v_{1}(x) − 1)^{2} > (v_{1}(x) + 1)^{2} + δ\}· \end{aligned}

If for all δ > 0 A_{δ}^{−} = ∅ then (v_{1}(x)−1)^{2} ≥ (v_{1}(x)+1)^{2} for every x ∈ (a, b) and we choose w_{n} ∉ ℚ with w_{n} → v_{2}. Then

F((v_{1}, v_{2}); (a, b)) ≤ lim inf_{n→∞} ∫_{a}^{b} f(v, w_{n}) dx = lim inf_{n→∞} ∫_{a}^{b}[(v_{1}(x) + 1)^{2} + (w_{n})^{2}] dx
≤ ∫_{a}^{b} ψ(v_{1}) dx + (v_{2})^{2}(b − a).

Similary (5.17) holds if for all δ > 0 A_{δ}^{+} = ∅. Thus assume that for δ > 0 sufficiently small A_{δ}^{−} ≠ ∅ ≠ A_{δ}^{+}. Choose a cut-off function φ_{δ} ∈ C^{∞}((a, b); [0, 1]) such that φ_{δ} ≡ 1 in A_{δ}^{−}, φ_{δ} ≡ 0 in A_{δ}^{+}, and let z_{n} ∈ ℚ, w_{n} ∉ ℚ, be such that with w_{n}, z_{n} → v_{2}. Define

v_{2}^{n,δ}(x) := φ_{δ}(x) z_{n} + (1 − φ_{δ}(x)) w_{n}.

Since (v_{2}^{n,δ})′ = φ′_{δ}(x)(z_{n} − w_{n}), it is clear that

lim_{δ→0^{+}} lim_{n→∞} ||v_{2}^{n,δ} − v_{2}||_{L^{2}} = lim_{δ→0^{+}} lim_{n→∞} ||(v_{2}^{n,δ})′||_{H^{−1}} = 0.

We have

F((v_{1}, v_{2}); (a, b)) ≤ lim inf_{δ→0^{+}} lim inf_{n→∞} ∫_{a}^{b} f(v_{1}, v_{2}^{n,δ}) dx

= lim inf_{δ→0^{+}} lim inf_{n→∞}{∫_{A_{δ}^{−}} f(v_{1}, z_{n}) dx + ∫_{A_{δ}^{+}} f(v_{1}, w_{n}) dx + ∫_{(a,b)\(A_{δ}^{−}∪A_{δ}^{+})} f(v_{1}^{n}, v_{2}^{n,δ}) dx}

≤ lim inf_{δ→0^{+}} lim inf_{n→∞}{∫_{A_{δ}^{−}}[ψ(v_{1}) + (z_{n})^{2}] dx + ∫_{A_{δ}^{+}}[ψ(v_{1}) + (w_{n})^{2}] dx + C |(a, b) \ (A_{δ}^{−} ∪ A_{δ}^{+})|}

= lim inf_{δ→0^{+}}{∫_{A_{δ}^{−}∪A_{δ}^{+}}[ψ(v_{1}) + (v_{2})^{2}] dx + C |(a, b) \ (A_{δ}^{−} ∪ A_{δ}^{+})|}

= ∫_{a}^{b} ψ(v_{1}) dx + (v_{2})^{2}(b − a),

where we have used the fact that

\begin{aligned}|(a, b) \ (A_{δ}^{−} ∪ A_{δ}^{+})| & = |\{x ∈ (a, b) : |(v_{1} − 1)^{2} − (v_{1} + 1)^{2}| < δ\}| = |\{x ∈ (a, b) : |v_{1}(x)| < δ/2\}| \\ & → |\{x ∈ (a, b) : v_{1}(x) = 0\}| = 0 \end{aligned}

as δ → 0^{+}.

The research of A. Braides was partially supported by the CNR research project “Equazioni Differenziali e Calcolo delle Variazioni”, that of I. Fonseca by the National Science Foundation through the Center for Nonlinear Analysis under Grant No. DMS–9731957, and that of G. Leoni by GNAFA, by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”, by the Italian CNR, through the strategic project “Metodi e modelli per la Matematica e l’Ingegneria”. G. Leoni would like to thank the hospitality and support of the Center for Nonlinear Analysis (NSF Grant No. DMS-9803791) and of S.I.S.S.A, where part of this work was carried out.

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