Uniform Regularity Estimates in Homogenization Theory of Elliptic System with Lower Order terms

TL;DR

This paper extends uniform regularity estimates for elliptic systems with oscillating coefficients, achieving sharp convergence rates and various regularity bounds without using traditional compactness methods.

Contribution

It develops new uniform regularity estimates for elliptic systems with lower order terms, using transformation techniques instead of compactness methods.

Findings

Established sharp $W^{1,p}$, Hölder, Lipschitz, and non-tangential maximal function estimates.

Achieved a sharp $O( ext{epsilon})$ convergence rate in $H_0^1( ext{Omega})$.

Extended regularity results to more general elliptic systems with oscillating periodic coefficients.

Abstract

In this paper, we extend the uniform regularity estimates obtained by M. Avellanda and F. Lin in the paper of Compactness methods in the theory of homogenization (Comm. Pure Appl. Math. 40(1987), no.6, 803-847) to the more general second order elliptic systems in divergences form, with rapidly oscillating periodic coefficients. We establish not only sharp $W^{1,p}$ estimates, Holder estimates, Lipschitz estimates and non-tangential maximal function estimates for the Dirichelt problem on a $C^{1,\eta}$ domain, but also a sharp $O(\varepsilon)$ convergence rate in $H_0^1(\Omega)$ by virtue of the Dirichlet correctors. The well known compactness methods are not employed here, instead we construct the transformations to make full use of the corresponding results developed by M. Avellanda and F. Lin.