Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients

Abstract

We study the asymptotic behaviour of the resolvents (Aε+I)−1 of elliptic second-order differential operators Aε in Rd with periodic rapidly oscillating coefficients, as the period 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 ε ) and the “double-porosity” case of coefficients that take contrasting values of order one and of order ε2 in different parts of the period cell. We provide a construction for the leading order term of the “operator asymptotics” of (Aε+I)−1 in the sense of operator-norm convergence and prove order O(ε) remainder estimates