Uniform boundary regularity in almost-periodic homogenization

TL;DR

This paper extends quantitative homogenization theory for elliptic systems with almost-periodic coefficients, establishing uniform boundary regularity estimates in various domain types, advancing understanding of boundary behavior in homogenized systems.

Contribution

It generalizes homogenization theory to almost-periodic coefficients and derives uniform boundary regularity estimates in complex domains.

Findings

Established large scale uniform boundary Lipschitz estimates for Dirichlet and Neumann problems.

Derived large scale uniform boundary Hölder estimates in $C^{1,eta}$ domains.

Proved $L^2$ Rellich estimates in Lipschitz domains.

Abstract

In the present paper, we generalize the theory of quantitative homogenization for second-order elliptic systems with rapidly oscillating coefficients in $APW^2(\mathbb{R}^d)$, which is the space of almost-periodic functions in the sense of H. Weyl. We obtain the large scale uniform boundary Lipschitz estimate, for both Dirichlet and Neumann problems in $C^{1,\alpha}$ domains. We also obtain large scale uniform boundary H\"{o}lder estimates in $C^{1,\alpha}$ domains and $L^2$ Rellich estimates in Lipschitz domains.