# Stochastic homogenization of maximal monotone relations and applications

## TL;DR

This paper develops a mathematical framework for understanding how random maximal monotone operators behave under homogenization, with applications to PDE systems in electromagnetism and elasticity.

## Contribution

It introduces a novel approach combining Fitzpatrick's variational formulation, Visintin's scale integration, and Tartar-Murat's compactness methods for stochastic homogenization.

## Key findings

- Established homogenization results for random maximal monotone operators.
- Applied the theory to PDE systems in electromagnetism and nonlinear elasticity.
- Provided a rigorous mathematical foundation for stochastic homogenization in complex systems.

## Abstract

We study the homogenization of a stationary random maximal monotone operator on a probability space equipped with an ergodic dynamical system. The proof relies on Fitzpatrick's variational formulation of monotone relations, on Visintin's scale integration/disintegration theory and on Tartar-Murat's compensated compactness. We provide applications to systems of PDEs with random coefficients arising in electromagnetism and in nonlinear elasticity.