Convergence Rates and Interior Estimates in Homogenization of Higher Order Elliptic Systems

TL;DR

This paper advances the understanding of homogenization for higher-order elliptic systems by establishing sharp convergence rates and interior regularity estimates, with applications to fundamental solutions.

Contribution

It provides the first sharp $O( ext{epsilon})$ convergence rate in $W^{m-1,p_0}$ and uniform interior regularity estimates for $2m$-order elliptic systems with oscillating coefficients.

Findings

Sharp $O( ext{epsilon})$ convergence rate in $W^{m-1,p_0}$

Uniform interior $C^{m-1,1}$ estimates

Asymptotic expansions for fundamental solutions

Abstract

This paper is concerned with the quantitative homogenization of $2m$-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp $O(\varepsilon)$ convergence rate in $W^{m-1, p_0}$ with $p_0=\frac{2d}{d-1}$ in a bounded Lipschitz domain in $\mathbb{R}^d$ as well as the uniform large-scale interior $C^{m-1, 1}$ estimate. With additional smoothness assumptions, the uniform interior $C^{m-1, 1}$, $W^{m,p}$ and $C^{m-1, \alpha}$ estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundamental solutions.