Uniform boundary regularity in almost-periodic homogenization

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.