Convergence rates in almost-periodic homogenization of higher-order elliptic systems

Article type: Research Article

Authors: Xu, Yao^a | Niu, Weisheng^{b; *}

Affiliations: [a] Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China. E-mail: xuyao89@gmail.com | [b] School of Mathematical Science, Anhui University, Hefei, 230601, China. E-mail: niuwsh@ahu.edu.cn

Correspondence: [*] Corresponding author. E-mail: niuwsh@ahu.edu.cn.

Abstract: This paper concentrates on the quantitative homogenization of higher-order elliptic systems with almost-periodic coefficients in bounded Lipschitz domains. For almost-periodic coefficients in the sense of H. Weyl, we establish uniform local L2 estimates for the approximate correctors. Under an additional assumption (1.8) on the frequencies of the coefficients, we derive the existence of true correctors as well as the O(ε) convergence rate in Hm−1. As a byproduct, the large-scale Hölder estimate and a Liouville theorem are obtained for higher-order elliptic systems with almost-periodic coefficients in the sense of Besicovitch. Since (1.8) is not well-defined for equivalence classes of almost-periodic functions in the sense of H. Weyl or Besicovitch, we provide another condition yielding the O(ε) convergence rate under perturbations of the coefficients.

Keywords: Almost periodic homogenization, higher-order elliptic systems, quantitative estimates

DOI: 10.3233/ASY-201627

Journal: Asymptotic Analysis, vol. 123, no. 1-2, pp. 95-137, 2021