Convergence Rates in Almost-Periodic Homogenization of Higher-order Elliptic Systems

TL;DR

This paper establishes quantitative convergence rates and corrector existence for higher-order elliptic systems with almost-periodic coefficients, advancing homogenization theory in complex bounded domains.

Contribution

It introduces new conditions for sharp $O( ext{epsilon})$ convergence rates and corrector existence in almost-periodic homogenization of higher-order elliptic systems.

Findings

Uniform local $L^2$ estimates for approximate correctors

Existence of true correctors under frequency assumptions

Sharp $O( ext{epsilon})$ convergence rate in $H^{m-1}$

Abstract

This paper concentrates on the quantitative homogenization of higher-order elliptic systems with almost-periodic coefficients in bounded Lipschitz domains. For coefficients which are almost-periodic in the sense of H. Weyl, we establish uniform ocal $L^2$ estimates for the approximate correctors. Under an additional assumption on the frequencies of the coefficients (see (1.10)), we derive the existence of the true correctors as well as the sharp $O(\varepsilon)$ convergence rate in $H^{m-1}$. As a byproduct, the large-scale H\"older estimate and a Liouville theorem are obtained for higher-order elliptic systems with almost-periodic coefficients in the sense of Besicovish. Since (1.10) is not well-defined for the equivalence classes of almost-periodic functions in the sense of H. Weyl or Besicovish, we provide another condition that implies the sharp convergence rate in terms of perturbations on the coefficients.