Convergence rates and Hölder estimates in almost-periodic homogenization of elliptic systems

Abstract

For a family of second-order elliptic systems in divergence form with rapidly oscillating almost-periodic coefficients, we obtain estimates for approximate correctors in terms of a function that quantifies the almost periodicity of the coefficients. The results are used to investigate the problem of convergence rates. We also establish uniform Hölder estimates for the Dirichlet problem in a bounded C 1,α domain.MSC2010: 35B27, 35J55.

Uniform boundary regularity in almost-periodic homogenization

“…This idea more or less has been shown in [20] for periodic homogenization, whose rate of convergence is always the same, i.e., ω 1,σ (ε) = O(ε). But it is of particular interest for almost-periodic homogenization since the rate of convergence could be arbitrarily slow.…”

“…It is obvious to see that the argument above for the large scale boundary Lipschitz estimate also works for the interior Lipschitz estimate; see [23, Theorem 11.1] for another proof. Indeed, we are able to establish 20) where u ε is a solution for L ε u ε + λu ε = F in B 2 .…”

“…The proof of Theorem 1.1 or 1.2 is roughly divided into three steps, which follow the same line as [20]:…”

“…In this section, we will extend the rate of convergence from C 1,1 domains to general Lipschitz domains for elliptic systems with almost-periodic coefficients. Our argument follows the same ideas as [20] and particularly relies on the solvability of L 2 elliptic boundary value problems with constant coefficients in Lipschitz domains. Now we state the main result of this section as follows: Theorem 3.1.…”

“…The Dirichlet and Neumann cases will be treated in two subsections separately. We modify the argument in [20] to make it adapted to general λ > 0.…”