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

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