Quantitative homogenization of differential forms

Abstract

We develop a quantitative theory of stochastic homogenization in a general framework involving differential forms. Inspired by recent progress in the uniformly elliptic setting, the analysis relies on the study of certain sub- and superadditive quantities. We establish an algebraic rate of convergence for these quantities and an algebraic error estimate for the homogenization of the Dirichlet problem. Most of the ideas needed in this article come from two distinct theories, the theory of quantitative stochastic homogenization, and the generalization of the main results of functional analysis and of the regularity theory of second-order elliptic equations to the setting of differential forms.