# Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems

## TL;DR

This paper establishes optimal homogenization error rates for nonlinear elliptic PDEs with random coefficients, improving upon previous algebraic convergence results and applicable to systems with regularity.

## Contribution

It provides the first optimal-order error estimates for stochastic homogenization of nonlinear elliptic equations and systems with stationary, mixing coefficients.

## Key findings

- Homogenization error of order ε for dimensions d≥3
- Error of order ε|log ε|^{1/2} in 2D
- Error estimates for representative volume approximation

## Abstract

We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on $\mathbb{R}^d$ with stationary law (i.e. spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale $\varepsilon>0$, we establish homogenization error estimates of the order $\varepsilon$ in case $d\geq 3$, respectively of the order $\varepsilon |\log \varepsilon|^{1/2}$ in case $d=2$. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence $\varepsilon^\delta$. We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order $(L/\varepsilon)^{-d/2}$ for a representative volume of size $L$. Our results also hold in the case of systems for which a (small-scale) $C^{1,\alpha}$ regularity theory is available.