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

TL;DR

This paper establishes the stochastic homogenisation (via $b3$-convergence) of random integral functionals constrained by differential operators with finite-dimensional nullspaces, extending previous results to symmetric gradients and deviatoric operators.

Contribution

It generalizes stochastic homogenisation results to a broader class of differential operators, including symmetric gradients and deviatoric operators, using a novel blow-up method and ergodic theorem.

Findings

Proves $b3$-convergence of random functionals with differential constraints.

Extends homogenisation results to symmetric gradients and deviatoric operators.

Employs a variant of the blow-up method and ergodic theorem for the analysis.

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.