Stochastic homogenization of degenerate integral functionals and their Euler-Lagrange equations

Abstract

We prove stochastic homogenization for integral functionals defined on Sobolev spaces, where the stationary, ergodic integrand satisfies a degenerate growth condition of the form c|ξA(ω,x)| p ≤f(ω,x,ξ)≤|ξA(ω,x)| p +Λ(ω,x) for some p∈(1,+∞) and with a stationary and ergodic diagonal matrix A such that its norm and the norm of its inverse satisfy minimal integrability assumptions and Λ is a nonnegative, stationary function with finite first moment. We also consider the convergence when Dirichlet boundary conditions or an obstacle condition are imposed. Assuming the strict convexity and differentiability of f with respect to its last variable, we further prove that the homogenized integrand is also strictly convex and differentiable. These properties allow us to show homogenization of the associated Euler-Lagrange equations.