Stochastic homogenization of subdifferential inclusions via scale integration

TL;DR

This paper develops a stochastic homogenization framework for subdifferential inclusions with convex random fields, demonstrating convergence to a deterministic system using ergodic theory and a novel scale integration method.

Contribution

It introduces a new scale integration approach for stochastic homogenization of subdifferential inclusions, extending previous methods to convex random fields with p-growth.

Findings

Solutions converge to a deterministic system with the same subdifferential structure.

The proof utilizes Birkhoff's ergodic theorem and maximal monotonicity.

A new scale integration technique is developed for this context.

Abstract

We study the stochastic homogenization of the system -div \sigma^\epsilon = f^\epsilon \sigma^\epsilon \in \partial \phi^\epsilon (\nabla u^\epsilon), where (\phi^\epsilon) is a sequence of convex stationary random fields, with p-growth. We prove that sequences of solutions (\sigma^\epsilon,u^\epsilon) converge to the solutions of a deterministic system having the same subdifferential structure. The proof relies on Birkhoff's ergodic theorem, on the maximal monotonicity of the subdifferential of a convex function, and on a new idea of scale integration, recently introduced by A. Visintin.