# Resolvent estimates for high-contrast elliptic problems with periodic coefficients

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20 Citations (SciVal)
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.