Homogenization of equi-coercive nonlinear energies defined on vector-valued functions, with non-uniformly bounded coefficients

Abstract

The present paper deals with the asymptotic behavior of equi-coercive sequences $\{{\mathcal{F}}_n\}$ of nonlinear functionals defined over vector-valued functions in $W_{)}^{1,p}(\Omega {)}^M$ , where $p>1$, $M\ge 1$, and $\Omega$ is a bounded open set of ${\mathbb{R}}^N$, $N\ge 2$. The strongly local energy density $F_n(\cdot ,Du)$ of the functional $\{{\mathcal{F}}_n\}$ satisfies a Lipschitz condition with respect to the second variable, which is controlled by a positive sequence $\{a_n\}$ which is only bounded in some suitable space $L^r(\Omega )$. We prove that the sequence $\{{\mathcal{F}}_n\}$ $\Gamma$-converges for the strong topology of $L^p(\Omega {)}^M$ to a functional $\mathcal{F}$ which has a strongly local density $F(\cdot ,Du)$ for sufficiently regular functions $u$. This compactness result extends former results on the topic, which are based either on maximum principle arguments in the nonlinear scalar case, or adapted div-curl lemmas in the linear case. Here, the vectorial character and the nonlinearity of the problem need a new approach based on a careful analysis of the asymptotic minimizers associated with the functional ${\mathcal{F}}_n$. The relevance of the conditions which are imposed to the energy density $F_n(\cdot ,Du)$, is illustrated by several examples including some classical hyper-elastic energies.