Quantitative Homogenization of Differential Forms

TL;DR

This paper extends stochastic homogenization theory to differential forms, providing algebraic convergence rates and error estimates, bridging elliptic PDE regularity and functional analysis in a new geometric framework.

Contribution

It introduces a quantitative homogenization framework for differential forms, combining stochastic analysis with advanced elliptic regularity techniques.

Findings

Established algebraic convergence rates for homogenization of differential forms.

Derived error estimates for Dirichlet problem homogenization.

Unified stochastic homogenization with differential forms and elliptic regularity theory.

Abstract

We develop a quantitative theory of stochastic homogenization in the more general framework of differential forms. Inspired by recent progress in the uniformly elliptic setting, the analysis relies on the study of certain subadditive quantities. We establish an algebraic rate of convergence from these quantities and deduce from this an algebraic error estimate for the homogenization of the Dirichlet problem. Most of the ideas needed in this article comes from two distinct theory, 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.