Resolvent estimates for high-contrast elliptic problems with periodic coefficients

TL;DR

This paper investigates the asymptotic behavior of resolvents of elliptic operators with periodic coefficients, including high-contrast cases, providing operator-norm approximations and error estimates as the period tends to zero.

Contribution

It introduces a unified approach to analyze resolvent asymptotics for both classical and high-contrast periodic elliptic operators, with explicit leading order terms and error bounds.

Findings

Derived operator-norm asymptotics for resolvents as period tends to zero.

Established order $O( ext{}\varepsilon)$ error estimates for the approximations.

Included high-contrast coefficients with contrasting values in the analysis.

Abstract

We study the asymptotic behaviour of the resolvents $({\mathcal A}^\varepsilon+I)^{-1}$ of elliptic second-order differential operators ${\mathcal A}^\varepsilon$ in ${\mathbb R}^d$ with periodic rapidly oscillating coefficients, as the period $\varepsilon$ goes to zero. The class of operators covered by our analysis includes both the "classical" case of uniformly elliptic families (where the ellipticity constant does not depend on $\varepsilon$) and the "double-porosity" case of coefficients that take contrasting values of order one and of order $\varepsilon^2$ in different parts of the period cell. We provide a construction for the leading order term of the "operator asymptotics" of $({\mathcal A}^\varepsilon+I)^{-1}$ in the sense of operator-norm convergence and prove order $O(\varepsilon)$ remainder estimates.