Uniform Estimates of Resolvents in Homogenization Theory of Elliptic Systems

TL;DR

This paper establishes uniform resolvent estimates for elliptic systems with periodic coefficients in homogenization theory, and analyzes their asymptotic behavior as the oscillation parameter tends to zero.

Contribution

It provides new uniform $L^p$ and Sobolev space resolvent estimates for elliptic operators with oscillating coefficients, advancing homogenization analysis.

Findings

Uniform $L^p$ resolvent estimates established

Asymptotic convergence of resolvents analyzed

Green function techniques used for behavior study

Abstract

In this paper, we study the estimates of resolvents $ R(\lambda,\mathcal{L}_{\varepsilon})=(\mathcal{L}_{\varepsilon}-\lambda I)^{-1} $, where $$ \mathcal{L}_{\varepsilon}=-\operatorname{div}(A(x/\varepsilon)\nabla) $$ is a family of second elliptic operators with symmetric, periodic and oscillating coefficients defined on a bounded domain $ \Omega $ with $ \varepsilon>0 $. For $ 1<p<\infty $, we will establish uniform $ L^p\to L^p $, $ L^p\to W_0^{1,p} $, $ W^{-1,p}\to L^p $ and $ W^{-1,p}\to W_0^{1,p} $ estimates by using the real variable method. Meanwhile, we use Green functions for operators $ \mathcal{L}_{\varepsilon}-\lambda I $ to study the asymptotic behavior of $ R(\lambda,\mathcal{L}_{\varepsilon}) $ and obtain convergence estimates in $ L^p\to L^p $, $ L^p\to W_0^{1,p} $ norm.