Stochastic homogenization of fully nonlinear uniformly elliptic equations revisited

TL;DR

This paper revisits stochastic homogenization for fully nonlinear elliptic equations, simplifying the obstacle problem approach and extending it to equations involving gradients, addressing previous estimation challenges.

Contribution

It provides a simplified proof of stochastic homogenization for nonlinear elliptic equations, including those with gradient dependence, using an improved obstacle problem method.

Findings

Simplified the obstacle problem approach for stochastic homogenization.

Extended the method to equations depending on the gradient.

Overcame previous difficulties with derivative estimates in correctors.

Abstract

We give a simplified presentation of the obstacle problem approach to stochastic homogenization for elliptic equations in nondivergence form. Our argument also applies to equations which depend on the gradient of the unknown function. In the latter case, we overcome difficulties caused by a lack of estimates for the first derivatives of approximate correctors by modifying the perturbed test function argument to take advantage of the spreading of the contact set.