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

We study stochastic homogenization for convex integral functionals|u bar right arrow integral(D) W(omega, x/epsilon, del u) dx, where u : D subset of R-d -> R-m,|defined on Sobolev spaces. Assuming only stochastic integrability of the map omega bar right arrow W (omega, 0, xi), we prove homogenization results under two different sets of assumptions, namely|center dot(1) W satisfies superlinear growth quantified by the stochastic integrability of the Fenchel conjugate W-*(center dot, 0, xi) and a certain monotonicity condition that ensures that the functional does not increase too much by componentwise truncation of u,|center dot(2) W is p-coercive in the sense |xi| (p) d - 1.|Condition center dot(2) directly improves upon earlier results, where p-coercivity with p > d is assumed and center dot(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.