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

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 ${\mathit{L}}^2$ 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 $\mathit{O}\mathrm{(}\mathit{\epsilon }\mathrm{)}$ convergence rate in ${\mathit{H}}^{\mathit{m}-1}$ . 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 $\mathit{O}\mathrm{(}\mathit{\epsilon }\mathrm{)}$ convergence rate under perturbations of the coefficients.