Archive for Rational Mechanics and Analysis

, Volume 219, Issue 3, pp 1061–1086 | Cite as

Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients

  • K. D. CherednichenkoEmail author
  • S. Cooper
Open Access


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.


Partial Differential Equation Principal Term Resolvent Estimate Double Porosity Homogenise Operator 
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Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK

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