Relaxation of p-growth integral functionals under space-dependent differential constraints

TL;DR

This paper derives a representation formula for the relaxed form of integral energies involving space-dependent differential constraints and p-growth conditions, expanding the understanding of variational problems with variable coefficients.

Contribution

It introduces a new relaxation formula for integral functionals with space-dependent constraints within the framework of -quasiconvexity, considering variable coefficients.

Findings

Derived a representation formula for relaxed integral energies.

Extended relaxation results to space-dependent differential constraints.

Applicable to functionals with p-growth conditions.

Abstract

A representation formula for the relaxation of integral energies $$(u,v)\mapsto\int_{\Omega} f(x,u(x),v(x))\,dx,$$ is obtained, where $f$ satisfies $p$-growth assumptions, $1<p<+\infty$, and the fields $v$ are subjected to space-dependent first order linear differential constraints in the framework of $\mathscr{A}$-quasiconvexity with variable coefficients.

Equations

$$(u,v)\mapsto {\int }_{\Omega }f(x,u(x),v(x))dx,$$

$$u\mapsto {\int }_{\Omega }f(x,u(x),\nabla u(x),\dots ,{\nabla }^{k}u(x))dx$$

$$(u,v)\mapsto {\int }_{\Omega }f(x,u(x),v(x))dx,$$

$$\mathcal{A}:L^p(\Omega ;{\mathbb{R}}^d)\to W^{-1,p}(\Omega ;{\mathbb{R}}^l),d,l\in \mathbb{N},$$

$$\mathcal{A}v:=\sum _{i=1}^N{A}^i(x)\frac{\partial v(x)}{\partial x_i}$$

$$\mathbb{A}(x,w):=\sum _{i=1}^N{A}^i(x)w_i\text{for }(x,w)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,$$

$$\text{rank }\mathbb{A}(x,w)=r\text{for every }x\in {\mathbb{R}}^N\text{and }w\in {\mathbb{S}}^{n-1}.$$

$$Q:=\Big{(}-\frac{1}{2},\frac{1}{2}\Big{)}.$$

$$\mathcal{C}_{x}:=\Big{\{}w\in C^{\infty}_{\rm per}(\mathbb{R}^{N};\mathbb{R}^{m}):\,\int_{Q}{w(y)\,dy}=0,\,\text{and }\sum_{i=1}^{N}A^{i}(x)\frac{\partial w(y)}{\partial y_{i}}=0\Big{\}}.$$

$$Q_{\mathscr{A}(x)}f(x,u,\xi):=\inf\Big{\{}\int_{Q}f(x,u,\xi+w(y))\,dy:\,w\in\mathcal{C}_{x}\Big{\}}.$$

$$A^i(x)\equiv A^i\in {\mathbb{M}}^{l\times d}\text{for every }x\in {\mathbb{R}}^N,i=1,\dots ,N.$$

$$I(u,v):={\int }_{\Omega }f(x,u(x),v(x))dx\text{for every }(u,v)\in L^p(\Omega ;{\mathbb{R}}^m)\times L^p(\Omega ;{\mathbb{R}}^d)$$

$$\text{(H)}0\le f(x,u,v)\le C(1+|u{|}^p+|v{|}^q),1<p,q<+\infty ,$$

$$\mathcal{I}((u,v),D):=\inf\Big{\{}\liminf_{n\to+\infty}\int_{D}f(x,u_{n}(x),v_{n}(x)):\,u_{n}\to u\quad\text{strongly in }L^{p}(\Omega;\mathbb{R}^{m}),\\ v_{n}\rightharpoonup v\quad\text{weakly in }L^{q}(\Omega;\mathbb{R}^{d})\,\text{and }\mathscr{A}v_{n}\to 0\quad\text{strongly in }W^{-1,q}(\Omega;\mathbb{R}^{l})\Big{\}}.$$

$${\int }_D{Q}_{\mathcal{A}(x)}f(x,u(x),v(x))dx=\mathcal{I}((u,v),D)$$

$$\frac{d\mathcal{I}((u,v)\cdot )(x)}{d{\mathcal{L}}^N}\le Q_{\mathcal{A}(x)}f(x,u(x),v(x))\text{for a.e. }x\in \Omega ,$$

$$\frac{1}{q}+\frac{1}{q^{\prime }}=1.$$

$$v(x+e_i)=v(x)i=1,\cdots ,N,$$

$$\mathcal{A}v:=\sum _{i=1}^N{A}^i(x)\frac{\partial v(x)}{\partial x_i}\in W^{-1,q}(\Omega ;{\mathbb{R}}^l).$$

$$\mathbb{A}(x,\lambda ):=\sum _{i=1}^N{A}^i(x){\lambda }_i\in {\mathbb{M}}^{l\times d}$$

$$\text{rank }\Big{(}\sum_{i=1}^{N}A^{i}(x)\lambda_{i}\Big{)}=r\quad\text{for all }x\in\mathbb{R}^{N}\text{ and }\lambda\in\mathbb{R}^{N}\setminus\{0\}.$$

$$\mathbb{Q}(x,\lambda )\mathbb{A}(x,\lambda )v:=v-\mathbb{P}(x,\lambda )v\text{for all }v\in {\mathbb{R}}^d,$$

$$\mathbb{Q}(x,\lambda )\xi =0\text{if }\xi \notin \text{Range }\mathbb{A}(x,\lambda ).$$

$${\mathbb{A}}_{\eta }(x,\lambda ):=\sum _{i=1}^N\eta (x)A^i(x){\lambda }_i,$$

$${\mathbb{P}}_{\eta }(x,\lambda ):={\eta }^2(x)\mathbb{P}(x,\lambda )\chi (|\lambda |)$$

$$\Vert P_{\eta }v{\Vert }_{L^q(\Omega ;{\mathbb{R}}^d)}\le C\Vert v{\Vert }_{L^q(\Omega ;{\mathbb{R}}^d)}$$

$$\Vert P_{\eta }v{\Vert }_{W^{-1,q}(\Omega ;{\mathbb{R}}^d)}\le C\Vert v{\Vert }_{W^{-1,q}(\Omega ;{\mathbb{R}}^d)},$$

$$\displaystyle\|v-P_{\eta}v\|_{L^{q}(\Omega;\mathbb{R}^{d})}\leq C\big{(}\|\mathscr{A}_{\eta}v\|_{W^{-1,q}(\Omega;\mathbb{R}^{l})}+\|v\|_{W^{-1,q}(\Omega;\mathbb{R}^{d})}\big{)},$$

$$\Vert {\mathcal{A}}_{\eta }{P}_{\eta }v{\Vert }_{W^{-1,q}(\Omega ;{\mathbb{R}}^l)}\le C\Vert v{\Vert }_{W^{-1,q}(\Omega ;{\mathbb{R}}^d)}$$

$$v_n\rightharpoonup v\text{weakly in }{L}^q(\Omega ;{\mathbb{R}}^d),$$

$$\mathcal{A}{v}_n\to 0\text{strongly in }{W}^{-1,q}(\Omega ;{\mathbb{R}}^l),$$

$$\{v_n\}\text{ generates the Young measure }\nu .$$

$$\mathcal{A}{\tilde{v}}_n\to 0\text{strongly in }{W}^{-1,s}(\Omega ;{\mathbb{R}}^l)\text{for every }1<s<q,$$

Full text

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is obtained, where $f$ satisfies $p$-growth assumptions, $1<p<+\infty$, and the fields $v$ are subjected to space-dependent first order linear differential constraints in the framework of $\mathscr{A}$-quasiconvexity with variable coefficients.

1. Introduction

The analysis of constrained relaxation problems is a central question in materials science. Many applications in continuum mechanics and, in particular, in magnetoelasticity, rely on the characterization of minimizers of non-convex multiple integrals of the type

[TABLE]

or

[TABLE]

where $\Omega$ is an open, bounded subset of $\mathbb{R}^{N}$, $u:\Omega\to\mathbb{R}^{m},\,m\in\mathbb{N}$, and the fields $v:\Omega\to\mathbb{R}^{d},\,d\in\mathbb{N}$, satisfy partial differential constraints of the type “$\mathscr{A}v=0$” other than ${\rm curl}\,v=0$ (see e.g. [5, 9]).

In this paper we provide a representation formula for the relaxation of non-convex integral energies of the form (1.1), in the case in which the energy density $f$ satisfies $p$-growth assumptions, and the fields $v$ are subjected to linear first-order space-dependent differential constraints.

The natural framework to study this family of relaxation problems is within the theory of $\mathscr{A}$-quasiconvexity with variable coefficients. In order to present this notion, we need to introduce some notation.

For $i=1\cdots,N$, let $A^{i}\in C^{\infty}(\mathbb{R}^{N};\mathbb{M}^{l\times d})\cap W^{1,\infty}(\mathbb{R}^{N};\mathbb{M}^{l\times d})$, let $1<p<+\infty$, and consider the differential operator

[TABLE]

defined as

[TABLE]

for every $v\in L^{p}(\Omega;\mathbb{R}^{d})$, where (1.2) is to be interpreted in the sense of distributions. Assume that the symbol $\mathbb{A}:\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{M}^{l\times d}$,

[TABLE]

satisfies the uniform constant rank condition (see [22])

[TABLE]

Let $Q$ be the unit cube in $\mathbb{R}^{N}$ with sides parallel to the coordinate axis, i.e.,

[TABLE]

Denote by $C^{\infty}_{\rm per}(\mathbb{R}^{N};\mathbb{R}^{m})$ the set of $\mathbb{R}^{m}$-valued smooth maps that are $Q$-periodic in $\mathbb{R}^{N}$, and for every $x\in\Omega$ consider the set

[TABLE]

Let $f:\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{d}\to[0,+\infty)$ be a Carathéodory function. The $\mathscr{A}-$quasiconvex envelope of $f(x,u,\cdot)$ for $x\in\Omega$ and $u\in\mathbb{R}^{m}$ is defined for $\xi\in\mathbb{R}^{d}$ as

[TABLE]

We say that $f$ is $\mathscr{A}-$quasiconvex if $f(x,u,\xi)=Q_{\mathscr{A}(x)}f(x,u,\xi)$ for a.e. $x\in\Omega$, and for all $u\in\mathbb{R}^{m}$ and $\xi\in\mathbb{R}^{d}$.

The notion of $\mathscr{A}$-quasiconvexity was first introduced by B. Dacorogna in [8], and extensively characterized in [17] by I. Fonseca and S. Müller for operators $\mathscr{A}$ defined as in (1.2), satisfying the constant rank condition (1.3), and having constant coefficients,

[TABLE]

In that paper the authors proved (see [17, Theorems 3.6 and 3.7 ]) that under $p$-growth assumptions on the energy density $f$, $\mathscr{A}$-quasiconvexity is necessary and sufficient for the lower-semicontinuity of integral functionals

[TABLE]

along sequences $(u^{n},v^{n})$ satisfying $u^{n}\to u$ in measure, $v^{n}\rightharpoonup v$ in $L^{p}(\Omega;\mathbb{R}^{d})$, and $\mathscr{A}v^{n}\to 0$ in $W^{-1,p}(\Omega)$. We remark that in the framework $\mathscr{A}={\rm curl}$, i.e., when $v^{n}=\nabla\phi^{n}$ for some $\phi^{n}\in W^{1,p}(\Omega;\mathbb{R}^{m})$, $d=n\times m$, $\mathscr{A}$-quasiconvexity reduces to Morrey’s notion of quasiconvexity.

The analysis of properties of $\mathscr{A}-$quasiconvexity for operators with constant coefficients was extended in the subsequent paper [6], where A. Braides, I. Fonseca and G. Leoni provided an integral representation formula for relaxation problems under $p$-growth assumptions on the energy density, and presented (via $\Gamma$-convergence) homogenization results for periodic integrands evaluated along $\mathscr{A}-$free fields. These homogenization results were later generalized in [13], where I. Fonseca and S. Krömer worked under weaker assumptions on the energy density $f$. In [19, 20], simultaneous homogenization and dimension reduction was studied in the framework of $\mathscr{A}$-quasiconvexity with constant coefficients. Oscillations and concentrations generated by $\mathscr{A}$-free mappings are the subject of [14]. Very recently an analysis of the case in which the energy density is nonpositive has been carried out in [18], and applications to the theory of compressible Euler systems have been studied in [7]. A parallel analysis for operators with constant coefficients and under linear growth assumptions for the energy density has been developed in [1, 4, 15, 21]. A very general characterization in this setting has been obtained in [2], following the new insight in [12].

The theory of $\mathscr{A}$-quasiconvexity for operators with variable coefficients has been characterized by P. Santos in [23]. Homogenization results in this setting have been obtained in [10] and [11].

This paper is devoted to proving a representation result for the relaxation of integral energies in the framework of $\mathscr{A}$-quasiconvexity with variable coefficients. To be precise, let $1<p,q<+\infty$, $d,m,l\in\mathbb{N}$, and consider a Carathéodory function $f:\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{d}\to[0,+\infty)$ satisfying

[TABLE]

for a.e. $x\in\Omega$, and all $(u,v)\in\mathbb{R}^{m}\times\mathbb{R}^{d}$, with $C>0$.

Denoting by $\mathcal{O}(\Omega)$ the collection of open subsets of $\Omega$, for every $D\in\mathcal{O}(\Omega)$, $u\in L^{p}(\Omega;\mathbb{R}^{m})$ and $v\in L^{q}(\Omega;\mathbb{R}^{d})$ with $\mathscr{A}v=0$, we define

[TABLE]

Our main result is the following.

Theorem 1.1**.**

Let $\mathscr{A}$ be a first order differential operator with variable coefficients, satisfying (1.3). Let $f:\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{d}\to[0,+\infty)$ be a Carathéodory function satisfying (H). Then,

[TABLE]

for all $D\in\mathcal{O}(\Omega),u\in L^{p}(\Omega;\mathbb{R}^{m})$ and $v\in L^{q}(\Omega;\mathbb{R}^{d})$ with $\mathscr{A}v=0$.

Adopting the “blow-up” method introduced in [16], the proof of the theorem consists in showing that the functional $\mathcal{I}((u,v),\cdot)$ is the trace of a Radon measure absolutely continuous with respect to the restriction of the Lebesgue measure $\mathcal{L}^{N}$ to $\Omega$, and proving that for a.e. $x\in\Omega$ the Radon-Nicodym derivative $\frac{d\mathcal{I}((u,v)\cdot)(x)}{d\mathcal{L}^{N}}$ coincides with the $\mathscr{A}-$quasiconvex envelope of $f$.

The arguments used are a combination of the ideas from [6, Theorem 1.1] and from [23]. The main difference with [6, Theorem 1.1], which reduces to our setting in the case in which the operator $\mathscr{A}$ has constant coefficients, is in the fact that while defining the operator $\mathcal{I}$ in (1.4) we can not work with exact solutions of the PDE, but instead we need to study sequences of asymptotically $\mathscr{A}-$vanishing fields. As pointed out in [23], in the case of variable coefficients the natural framework is the context of pseudo-differential operators. In this setting, we don’t know how to project directly onto the kernel of the differential constraint, but we are able to construct an “approximate” projection operator $P$ such that for every field $v\in L^{p}$, the $W^{-1,p}$ norm of $\mathscr{A}Pv$ is controlled by the $W^{-1,p}$ norm of $v$ itself (we refer to [23, Subsection 2.1] for a detailed explanation of this issue and to the references therein for a treatment of the main properties of pseudo-differential operators). For the same reason, in the proof of the inequality

[TABLE]

an equi-integrability argument is needed (see Proposition 3.2). We also point out that the representation formula in Theorem 1.1 was obtained in a simplified setting in [11] as a corollary of the main homogenization result. Here we provide an alternative, direct proof, which does not rely on homogenization techniques.

The paper is organized as follows: in Section 2 we establish the main assumptions on the differential operator $\mathscr{A}$ and we recall some preliminary results on $\mathscr{A}-$quasiconvexity with variable coefficients. Section 3 is devoted to the proof of Theorem 1.1.

**Notation

**Throughout the paper $\Omega\subset\mathbb{R}^{N}$ is a bounded open set, $1<p,q<+\infty$, $\mathcal{O}(\Omega)$ is the set of open subsets of $\Omega$, $Q$ denotes the unit cube in $\mathbb{R}^{N}$, $Q(x_{0},r)$ and $B(x_{0},r)$ are, respectively, the open cube and the open ball in $\mathbb{R}^{N}$, with center $x_{0}$ and radius $r$. Given an exponent $1<q<+\infty$, we denote by $q^{\prime}$ its conjugate exponent, i.e., $q^{\prime}\in(1,+\infty)$ is such that

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Whenever a map $v\in L^{q},C^{\infty},\cdots$ is $Q-$periodic, that is

[TABLE]

for a.e. $x\in\mathbb{R}^{N}$, $\{e_{1},\cdots,e_{N}\}$ being the standard basis of $\mathbb{R}^{N}$, we write $v\in L^{q}_{\rm per},C^{\infty}_{\rm per},\dots$ We implicitly identify the spaces $L^{q}(Q)$ and $L^{q}_{\rm per}(\mathbb{R}^{N})$.

We adopt the convention that $C$ will denote a generic constant, whose value may change from line to line in the same formula.

2. Preliminary results

In this section we introduce the main assumptions on the differential operator $\mathscr{A}$ and we recall some preliminary results about $\mathscr{A}-$quasiconvexity.

For $i=1,\cdots,N$, $x\in\mathbb{R}^{N}$, consider the linear operators $A^{i}(x)\in\mathbb{M}^{l\times d}$, with $A^{i}\in C^{\infty}(\mathbb{R}^{N};\mathbb{M}^{l\times d})\cap W^{1,\infty}(\mathbb{R}^{N};\mathbb{M}^{l\times d})$. For every $v\in L^{q}(\Omega;\mathbb{R}^{d})$ we set

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The symbol $\mathbb{A}:\mathbb{R}^{N}\times\mathbb{R}^{N}\setminus\{0\}\to\mathbb{M}^{l\times d}$ associated to the differential operator $\mathscr{A}$ is

[TABLE]

for every $x\in\mathbb{R}^{N}$, $\lambda\in\mathbb{R}^{N}\setminus\{0\}$. We assume that $\mathscr{A}$ satisfies the following uniform constant rank condition:

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For every $x\in\mathbb{R}^{N}$, $\lambda\in\mathbb{R}^{N}\setminus\{0\}$, let $\mathbb{P}(x,\lambda):\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the linear projection on Ker $\mathbb{A}(x,\lambda)$, and let $\mathbb{Q}(x,\lambda):\mathbb{R}^{l}\to\mathbb{R}^{d}$ be the linear operator given by

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The main properties of $\mathbb{P}(\cdot,\cdot)$ and $\mathbb{Q}(\cdot,\cdot)$ are recalled in the following proposition (see e.g. [23, Subsection 2.1]).

Proposition 2.1**.**

Under the constant rank condition (2.1), for every $x\in\mathbb{R}^{N}$ the operators $\mathbb{P}(x,\cdot)$ and $\mathbb{Q}(x,\cdot)$ are, respectively, [math]-homogeneous and $(-1)$-homogeneous. In addition, $\mathbb{P}\in$ $C^{\infty}(\mathbb{R}^{N}\times\mathbb{R}^{N}\setminus\{0\};\mathbb{M}^{d\times d})$ and $\mathbb{Q}\in C^{\infty}(\mathbb{R}^{N}\times\mathbb{R}^{N}\setminus\{0\};\mathbb{M}^{d\times l})$.

Let $\eta\in C^{\infty}_{c}(\Omega;[0,1])$, $\eta=1$ in ${\Omega^{\prime}}$ for some ${\Omega^{\prime}}\subset\subset\Omega$. We denote by $\mathbb{A}_{\eta}$ the symbol

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for every $x\in\mathbb{R}^{N}$, $\lambda\in\mathbb{R}^{N}\setminus\{0\}$, and by $\mathscr{A}_{\eta}$ the corresponding pseudo-differential operator (see [23, Subsection 2.1] for an overview of the main properties of pseudo-differential operators). Let $\chi\in C^{\infty}(\mathbb{R}^{+};\mathbb{R})$ be such that $\chi(|\lambda|)=0$ for $|\lambda|<1$ and $\chi(|\lambda|)=1$ for $|\lambda|>2$. Let also $P_{\eta}$ be the operator associated to the symbol

[TABLE]

for every $x\in\mathbb{R}^{N}$, $\lambda\in\mathbb{R}^{N}\setminus\{0\}$. The following proposition (see [23, Theorem 2.2 and Subsection 2.1]) collects the main properties of the operators $P_{\eta}$ and $\mathscr{A}_{\eta}$.

Proposition 2.2**.**

Let $1<q<+\infty$, and let $\mathscr{A}_{\eta}$ and $P_{\eta}$ be the pseudo-differential operators associated with the symbols (2.2) and (2.3), respectively. Then there exists a constant $C$ such that

[TABLE]

for every $v\in L^{q}(\Omega;\mathbb{R}^{d})$, and

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for every $v\in W^{-1,q}(\Omega;\mathbb{R}^{d})$.

3. Proof of Theorem 1.1

Before proving Theorem 1.1 we state and prove a decomposition lemma, which generalizes [17, Lemma 2.15] to the case of operators with variable coefficients.

Lemma 3.1**.**

Let $1<q<+\infty$. Let $\mathscr{A}$ be a first order differential operator with variable coefficients, satisfying (2.1). Let $v\in L^{q}(\Omega;\mathbb{R}^{d})$, and let $\{v_{n}\}$ be a bounded sequence in $L^{q}(\Omega;\mathbb{R}^{d})$ such that

[TABLE]

Then, there exists a $q$-equiintegrable sequence $\{\tilde{v}_{n}\}\subset L^{q}(\Omega;\mathbb{R}^{d})$ such that

[TABLE]

In addition, if $\Omega\subset Q$ then we can construct the sequence $\{\tilde{v}^{n}\}$ so that $\tilde{v}_{n}-v\in L^{q}_{\text{per}}(\mathbb{R}^{N};\mathbb{R}^{d})$ for every $n\in\mathbb{N}$.

Proof.

Arguing as in the first part of [23, Proof of Theorem 1.1], we construct a $q$-equiintegrable sequence $\{\hat{v}_{n}\}$ satisfying (3.1), (3.2) and (3.3). The conclusion follows by setting $\tilde{v}_{n}:=\hat{v}_{n}-\int_{\Omega}\hat{v}_{n}(x)\,dx+\int_{\Omega}v(x)\,dx$.

In the case in which $\Omega\subset Q$, let $\{\varphi^{i}\}$ be a sequence of cut-off functions in $Q$ with $0\leq\varphi^{i}\leq 1$ in $Q$, such that $\varphi^{i}=0$ on $Q\setminus\Omega$ and $\varphi^{i}\to 1$ pointwise in $\Omega$. Define $w^{i}_{n}:=\varphi^{i}(\hat{v}_{n}-v)$. By (3.3) for every $\psi\in L^{q^{\prime}}(\Omega;\mathbb{R}^{d})$ we have

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By (3.1), (3.2), and the compact embedding of $L^{q}(\Omega;\mathbb{R}^{d})$ into $W^{-1,q}(\Omega;\mathbb{R}^{d})$, there holds

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as $n\to+\infty$, for every $1<s<q$. Extending the maps $w^{i}_{n}$ outside $Q$ by periodicity, by the metrizability of the weak topology on bounded sets and by Attouch’s diagonalization lemma (see [3, Lemma 1.15 and Corollary 1.16]), we obtain a sequence

[TABLE]

with $\{w_{n}\}\subset L^{q}_{\text{per}}(\mathbb{R}^{N};\mathbb{R}^{d})$, and such that $w_{n}+v$ satisfies (3.1), (3.2) and (3.3). The thesis follows by setting

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∎

The following proposition will allow us to neglect vanishing perturbations of $q$-equiintegrable sequences.

Proposition 3.2**.**

For every $n\in\mathbb{N}$, let $f_{n}:Q\times\mathbb{R}^{d}\to[0,+\infty)$ be a continuous function. Assume that there exists a constant $C>0$ such that, for $q>1$,

[TABLE]

and that the sequence $\{f_{n}(y,\cdot)\}$ is equicontinuous in $\mathbb{R}^{d}$, uniformly in $y$. Let $\{w_{n}\}$ be a $q$-equiintegrable sequence in $L^{q}(Q;\mathbb{R}^{d})$, and let $\{v_{n}\}\subset L^{q}(Q;\mathbb{R}^{d})$ be such that

[TABLE]

Then

[TABLE]

Proof.

Fix $\eta>0$. In view of (3.5), the sequence $\{C(1+|v_{n}|^{q}+|w_{n}|^{q})\}$ is equiintegrable in $Q$, thus there exists $0<\varepsilon<\frac{\eta}{3}$ such that

[TABLE]

for every $A\subset Q$ with $|A|<\varepsilon$. By the $q$-equiintegrability of $\{w_{n}\}$ and $\{v_{n}\}$, and by Chebyshev’s inequality there holds

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for every $n\in\mathbb{N}$. Therefore, there exists $M_{0}$ satisfying

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By the uniform equicontinuity of the sequence $\{f_{n}(y,\cdot)\}$, there exists $\delta>0$ such that, for every $\xi_{1},\xi_{2}\in\overline{B(0,M_{0})}$, with $|\xi_{1}-\xi_{2}|<\delta$, we have

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for every $n\in\mathbb{N}$. By (3.5) and Egoroff’s theorem, there exists a set $E_{\varepsilon}\subset Q$, $|E_{\varepsilon}|<\frac{\varepsilon}{2}$, such that

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and, in particular,

[TABLE]

for every $n\geq n_{0}$, for some $n_{0}\in\mathbb{N}$.

We observe that

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The first term in the right-hand side of (3.10) can be further decomposed as

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We observe that by (3.7)

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Hence, for $n\geq n_{0}$, by (3.4), (3.6), (3.8), and (3.9) we deduce the estimate

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The thesis follows by the arbitrariness of $\eta$. ∎

We now prove our main result.

Proof of Theorem 1.1.

The proof is subdivided into 4 steps. Steps 1 and 2 follow along the lines of [6, Proof of Theorem 1.1]. Step 3 is obtained by modifying [6, Lemma 3.5], whereas Step 4 follows by adapting an argument in [23, Proof of Theorem 1.2]. We only outline the main ideas of Steps 1 and 2 for convenience of the reader, whilst we provide more details for Steps 3 and 4.

Step 1:

The first step consists in showing that

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This identification is proved by adapting [6, Proof of Lemma 3.1]. The only difference is the application of Lemma 3.1 instead of [6, Proposition 2.3 (i)].

Step 2:

The second step is the proof that $\mathcal{I}((u,v),\cdot)$ is the trace of a Radon measure absolutely continuous with respect to $\mathcal{L}^{N}\lfloor\Omega$. This follows as a straightforward adaptation of [6, Lemma 3.4]. The only modifications are due to the fact that [6, Proposition 2.3 (i)] and [6, Lemma 3.1] are now replaced by Lemma 3.1 and Step 1.

Step 3:

We claim that

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Indeed, since $g(x,\xi):=f(x,u(x),\xi)$ is a Carathéodory function, by Scorza-Dragoni Theorem there exists a sequence of compact sets $K_{j}\subset\Omega$ such that

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and the restriction of $g$ to $K_{j}\times\mathbb{R}^{d}$ is continuous. Hence, the set

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where $K_{j}^{*}$ is the set of Lebesgue point for the characteristic function of $K_{j}$ and $\mathcal{L}(u,v)$ is the set of Lebesgue points of $u$ and $v$, is such that

[TABLE]

and so $|\Omega\setminus\omega|=0$. Let $x_{0}\in\omega$ be such that

[TABLE]

and

[TABLE]

where the sequence of radii $r$ is such that $\mathcal{I}((u,v),\partial Q(x_{0},r))=0$ for every $r$. (Such a choice of the sequence is possible due to Step 2).

By Step 1, for every $r$ there exists a $q-$equiintegrable sequence $\{v_{n,r}\}$ such that

[TABLE]

as $n\to+\infty$, and

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A change of variables yields

[TABLE]

where

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Arguing as in [6, Proof of Lemma 3.5], Hölder’s inequality and a change of variables imply

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as $n\to+\infty$ and $r\to 0^{+}$, in this order. We claim that

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as $n\to+\infty$, for every $r$ and every $1<s<q$.

Indeed, let $\varphi\in W^{1,s^{\prime}}_{0}(Q;\mathbb{R}^{d})$. There holds

[TABLE]

where \psi_{r}(x):=\varphi\big{(}\frac{x-x_{0}}{r}\big{)} for a.e. $x\in Q(x_{0},r)$. Since $\psi_{r}\in W^{1,s^{\prime}}_{0}(Q(x_{0},r);\mathbb{R}^{d})$ and

[TABLE]

we obtain the estimate

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Claim (3.18) follows by (3.16).

In view of (3.17) and (3.18), a diagonalization procedure yields a $q-$equiintegrable sequence $\{\hat{w}_{k}\}\subset L^{q}(Q;\mathbb{R}^{d})$ satisfying

[TABLE]

and

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For every $\varphi\in W^{1,s^{\prime}}_{0}(Q;\mathbb{R}^{l})$, $1<s<q$, there holds

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Thus,

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for every $1<s<q$. By (3.19) and (3.20) we conclude that

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In view of (3.19) and (3.22), an adaptation of [6, Corollary 3.3] yields a $q-$equiintegrable sequence $\{w_{k}\}$ such that

[TABLE]

and

[TABLE]

Finally, by combining (3.21), (3.23), and (3.24), and by the definition of $\mathscr{A}$-quasiconvex envelope for operators with constant coefficients, we obtain

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for a.e. $x_{0}\in\Omega$. This concludes the proof of Claim (3.12).

Step 4:

To complete the proof of the theorem we need to show that

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To this aim, let $\mu>0$, and $x_{0}\in\omega$ be such that (3.14) and (3.15) hold. Let $w\in C^{\infty}_{\rm per}(\mathbb{R}^{N};\mathbb{R}^{d})$ be such that

[TABLE]

and

[TABLE]

Let $\eta\in C^{\infty}_{c}(\Omega;[0,1])$ be such that $\eta\equiv 1$ in a neighborhood of $x_{0}$ and let $r$ be small enough so that

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Consider a map $\varphi\in C^{\infty}_{c}(Q(x_{0},r);[0,1])$ satisfying

[TABLE]

and define

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We observe that $z^{r}_{m}\in L^{q}(\Omega;\mathbb{R}^{d})$, and for $\psi\in L^{q^{\prime}}(\Omega;\mathbb{R}^{d})$ we have

[TABLE]

By (3.26) and by the Riemann-Lebesgue lemma we have

[TABLE]

as $m\to+\infty$. We claim that

[TABLE]

where $\mathscr{A}_{\eta}$ is the pseudo-differential operator defined in (2.2). Indeed, by (3.28) we obtain

[TABLE]

By the regularity of the operators $A^{i}$ and by a change of variables, the first term in the right-hand side of (3.33) is estimated as

[TABLE]

In view of (3.26) the second term in the right-hand side of (3.33) becomes

[TABLE]

and thus converges to zero weakly in $L^{q}(\Omega;\mathbb{R}^{l})$, as $m\to+\infty$, due to (3.26) and by the Riemann-Lebesgue lemma. Hence,

[TABLE]

by the compact embedding of $L^{q}(\Omega;\mathbb{R}^{l})$ into $W^{-1,q}(\Omega;\mathbb{R}^{l})$. Finally, the third term in the right-hand side of (3.33) satisfies

[TABLE]

which again converges to zero weakly in $L^{q}(\Omega;\mathbb{R}^{l})$, as $m\to+\infty$, owing again to (3.26) and the Riemann-Lebesgue lemma. Therefore,

[TABLE]

Claim (3.32) follows by combining (3.34)–(3.36).

Consider the maps

[TABLE]

where $P_{\eta}$ is the projection operator introduced in (2.3). By Proposition 2.2 we have

[TABLE]

By (3.31) and (3.37), the sequence $\{v^{r}_{m}\}$ is uniformly bounded in $L^{q}(Q(x_{0},r);\mathbb{R}^{d})$. Thus, there exists a map $v^{r}\in L^{q}(Q(x_{0},r);\mathbb{R}^{d})$ such that, up to the extraction of a (not relabelled) subsequence,

[TABLE]

as $m\to+\infty$. Again by (3.31), and by the compact embedding of $L^{q}$ into $W^{-1,q}$, we deduce that

[TABLE]

as $m\to+\infty$. Therefore, by combining (3.38) and (3.41), we conclude that

[TABLE]

as $m\to+\infty$, and the convergence holds for the entire sequence. Additionally, by (3.28), (3.39), and (3.42), we obtain

[TABLE]

as $m\to+\infty$. Finally, by (3.32), (3.40), and (3.42), there holds

[TABLE]

We recall that, since $x_{0}$ satisfies (3.15), Step 1 yields

[TABLE]

We claim that

[TABLE]

where $g$ is the function introduced in Step 3. Indeed, for every $r\in\mathbb{R}$, consider the function $g^{r}:Q\times\mathbb{R}^{d}\to[0,+\infty)$ defined as

[TABLE]

Since $x_{0}\in\omega$, by (3.13) there exists $K_{j}$ such that $x_{0}\in K_{j}$. In particular, this yields the existence of $r_{0}>0$ such that for $r\leq r_{0}$, the maps $g^{r}$ are continuous on $Q\times\mathbb{R}^{d}$, and the family $\{g^{r}(y,\cdot)\}$ is equicontinuous in $\mathbb{R}^{d}$, uniformly with respect to $y$. A change of variables yields

[TABLE]

On the other hand, by (3.43) we have

[TABLE]

Therefore, by a diagonal procedure we extract a subsequence $\{m_{r}\}$ such that

[TABLE]

and

[TABLE]

In view of (3.14), (3.30) and the Riemann-Lebesgue lemma, the sequence $\{v(x_{0}+r\cdot)+z^{r}_{m_{r}}(x_{0}+r\cdot)\}$ is $q$-equiintegrable in $Q$. Hence, by (H) we are under the assumptions of Proposition 3.2, and we conclude that

[TABLE]

Claim (3.45) follows by combining (3.46) with (3.47).

Arguing as in [6, Proof of Lemma 3.5], for every $x_{0}\in\omega$ (where $\omega$ is the set defined in (3.13)) we have

[TABLE]

hence by (3.45) we deduce that

[TABLE]

By (3.30) we obtain

[TABLE]

The growth assumption (H) and estimate (3.29) yield

[TABLE]

Thus, by (3.48), the periodicity of $w$, and Riemann-Lebesgue lemma, we deduce

[TABLE]

where the last inequality is due to (3.27). Letting $\mu\to 0^{+}$ we conclude (3.25). ∎

Acknowledgements

The authors thank the Center for Nonlinear Analysis (NSF Grant No. DMS-0635983), where this research was carried out, and also acknowledge support of the National Science Foundation under the PIRE Grant No. OISE-0967140. The research of I. Fonseca and E. Davoli was funded by the National Science Foundation under Grant No. DMS- 0905778. E. Davoli acknowledges the support of the Austrian Science Fund (FWF) projects P 27052 and I 2375. The research of I. Fonseca was further partially supported by the National Science Foundation under Grant No. DMS-1411646.