Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients

Abstract

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