# Resolvent estimates for high-contrast elliptic problems with periodic coefficients

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### 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 $${\varepsilon^2}$$ 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.
Language English 1061-1086 Archive for Rational Mechanics and Analysis 219 3 7 Sep 2015 10.1007/s00205-015-0916-4 Published - Mar 2016

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Resolvent Estimates
Periodic Coefficients
Elliptic Problems
Porosity
Oscillating Coefficients
Operator Norm
Ellipticity
Operator
Remainder
Resolvent
Differential operator
Asymptotic Behavior
Cell
Zero
Coefficient
Term
Estimate

### Cite this

In: Archive for Rational Mechanics and Analysis, Vol. 219, No. 3, 03.2016, p. 1061-1086.

Research output: Contribution to journalArticle

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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 $${\varepsilon^2}$$ 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.",
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AU - Cooper,S.

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AB - 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.

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