Quantitative nonlinear homogenization: control of oscillations

TL;DR

This paper advances the understanding of nonlinear stochastic homogenization by establishing a quantitative two-scale expansion for genuinely nonlinear elliptic equations with random coefficients in low dimensions.

Contribution

It introduces a novel analysis of the linearized operator in nonlinear homogenization, enabling the first quantitative results for nonlinear elliptic systems with random coefficients.

Findings

Established annealed Meyers' estimates for the linearized operator

Derived optimal quantitative two-scale expansion results in low dimensions

Extended homogenization techniques to nonlinear elliptic equations with non-symmetric coefficients

Abstract

Quantitative stochastic homogenization of linear elliptic operators is by now well-understood. In this contribution we move forward to the nonlinear setting of monotone operators with $p$-growth. This work is dedicated to a quantitative two-scale expansion result. By treating the range of exponents $2\le p <\infty$ in dimensions $d\le 3$, we are able to consider genuinely nonlinear elliptic equations and systems such as $-\nabla \cdot A(x)(1+|\nabla u|^{p-2})\nabla u=f$ (with $A$ random, non-necessarily symmetric) for the first time. When going from $p=2$ to $p>2$, the main difficulty is to analyze the associated linearized operator, whose coefficients are degenerate, unbounded, and depend on the random input $A$ via the solution of a nonlinear equation. One of our main achievements is the control of this intricate nonlinear dependence, leading to annealed Meyers' estimates for the linearized operator, which are key to the optimal quantitative two-scale expansion result we derive (this is also new in the periodic setting).