Resolvent estimates for high-contrast elliptic problems with periodic coefficients

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.