Quantitative stochastic homogenization of convex integral functionals

TL;DR

This paper provides quantitative error estimates for the homogenization of convex integral functionals with random coefficients, achieving optimal stochastic integrability and applying results to local minimizers.

Contribution

It introduces an algebraic error estimate for the Dirichlet problem in stochastic homogenization of convex functionals, with optimal stochastic integrability.

Findings

Algebraic error estimate for homogenization error

Optimal stochastic integrability results

Quenched $C^{0,1}$ estimates for minimizers

Abstract

We present quantitative results for the homogenization of uniformly convex integral functionals with random coefficients under independence assumptions. The main result is an error estimate for the Dirichlet problem which is algebraic (but sub-optimal) in the size of the error, but optimal in stochastic integrability. As an application, we obtain quenched $C^{0,1}$ estimates for local minimizers of such energy functionals.