New homogenization results for convex integral functionals and their Euler–Lagrange equations

We study stochastic homogenization for convex integral functionals $$\begin{aligned} u\mapsto \int _D W(\omega ,\tfrac{x}{\varepsilon },\nabla u)\,\textrm{d}x,\quad \text{ where }\quad u:D\subset {\mathbb {R}}^d\rightarrow {\mathbb {R}}^m, \end{aligned}$$ defined on Sobolev spaces. Assuming only stochastic integrability of the map $$\omega \mapsto W(\omega ,0,\xi )$$ , we prove homogenization results under two different sets of assumptions, namely Condition $$\bullet _2$$ directly improves upon earlier results, where p-coercivity with $$p>d$$ is assumed and $$\bullet _1$$ provides an alternative condition under very weak coercivity assumptions and additional structure conditions on the integrand. We also study the corresponding Euler–Lagrange equations in the setting of Sobolev-Orlicz spaces. In particular, if $$W(\omega ,x,\xi )$$ is comparable to $$W(\omega ,x,-\xi )$$ in a suitable sense, we show that the homogenized integrand is differentiable.