Published Paper

Inserted: 28 mar 2023
Last Updated: 12 feb 2024

Journal: Calc. Var. Partial Differential Equations
Volume: 63
Pages: art. 32
Year: 2024
Doi: https://doi.org/10.1007/s00526-023-02636-x

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.