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Resolvent estimates for high-contrast elliptic problems with periodic coefficients (1404.5342v1)

Published 21 Apr 2014 in math.AP

Abstract: 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 $\varepsilon2$ 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.

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