Stochastic Homogenisation of nonconvex functionals in the space of $\mathbb{A}$-weakly differentiable maps

Abstract

We prove the $\Gamma$-convergence of sequences of differentially constrained, random integral functionals of the form \begin{equation*} \int_{U} f\Big(\omega, x/\varepsilon, \mathbb{A} u\Big) \mathrm{d} x \end{equation*} for the class of vectorial differential operators $\mathbb{A}$ with finite-dimensional nullspaces. This work is intended to generalise results for the full gradient and to cover the cases of symmetric gradients and the deviatoric operator. The homogenisation procedure is carried out by employing a variant of the blow-up method in the setting of $\mathbb{A}$-weakly differentiable maps along with the Akcloglu-Krengel subadditive ergodic theorem.