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

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 $O(\epsilon )$ 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.

37 pages

References in corpus (1)

• Homogenization of the Neumann problem for higher-order elliptic equations with periodic coefficients