Bounded correctors in almost periodic homogenization

TL;DR

This paper proves the existence of bounded, almost periodic correctors for linear elliptic equations with almost periodic coefficients, using a new quantification condition and advanced ergodic and regularity theories.

Contribution

Introduction of a new condition that quantifies almost periodicity, enabling the proof of bounded correctors in homogenization of elliptic equations.

Findings

Bounded, almost periodic correctors are established under the new condition.

The approach combines a quantitative ergodic theorem with recent regularity results.

Results include control over spatial averages of corrector gradients.

Abstract

We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov. The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by the first author and Shen for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincar\'e-type inequality.