Resolvent estimates for high-contrast elliptic problems with periodic coefficients

Abstract

We study the asymptotic behaviour of the resolvents $({\mathcal{A}}^{\epsilon }+I{)}^{-1}$ of elliptic second-order differential operators ${\mathcal{A}}^{\epsilon }$ in ${\mathbb{R}}^d$ with periodic rapidly oscillating coefficients, as the period $\epsilon$ 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 $\epsilon$) and the "double-porosity" case of coefficients that take contrasting values of order one and of order ${\epsilon }^2$ in different parts of the period cell. We provide a construction for the leading order term of the "operator asymptotics" of $({\mathcal{A}}^{\epsilon }+I{)}^{-1}$ in the sense of operator-norm convergence and prove order $O(\epsilon )$ remainder estimates.

20 pages

References in corpus (2)

• Interior error estimate for periodic homogenization
• Homogenization of the elliptic Dirichlet problem: operator error estimates in $L_2$

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