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

Abstract

We study stochastic homogenization for convex integral functionals $$u\mapsto \int_D W(ω,\tfrac{x}\varepsilon,\nabla u)\,\mathrm{d}x,\quad\mbox{where}\quad u:D\subset \mathbb{R}^d\to\mathbb{R}^m,$$ 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 ${\bullet }_1$ $W$ satisfies superlinear growth quantified by the stochastic integrability of the Fenchel conjugate $W^{\ast }(\cdot ,0,\xi )$ and a mild monotonicity condition that ensures that the functional does not increase too much by componentwise truncation of $u$, ${\bullet }_2$ $W$ is $p$-coercive in the sense $|\xi {|}^p\le W(\omega ,x,\xi )$ 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.

43 pages