*preprint*

**Inserted:** 28 mar 2023

**Year:** 2023

**Abstract:**

We study stochastic homogenization for convex integral functionals $$u\mapsto
\int_{D} W(\omega,\tfrac{x}\varepsilon,\nabla
u)\,\mathrm{d}x,\quad\mbox{where}\quad u:D\subset
\mathbb{R}^{d\to\mathbb{R}}^{m,$$} 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
$\bullet_1\quad$ $W$ satisfies superlinear growth quantified by the
stochastic integrability of the Fenchel conjugate $W^*(\cdot,0,\xi)$ and a mild
monotonicity condition that ensures that the functional does not increase too
much by componentwise truncation of $u$,
$\bullet_2\quad$ $W$ is $p$-coercive in the sense $

\xi

^p\leq
W(\omega,x,\xi)$ for some $p>d-1$.
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.