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.