Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients

By Kirill D. Cherednichenko, Shane Cooper. 2014

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We study the asymptotic behaviour of the resolvents $${(\mathcal{A}^\varepsilon+I)^{-1}}$$(Aε+I)-1 of elliptic second-order differential operators $${{\mathcal{A}}^\varepsilon}$$Aε in $${\mathbb{R}^d}$$Rd with periodic rapidly oscillating coefficients, as the period $${\varepsilon}$$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 $${\varepsilon}$$ε) and the “double-porosity” case of coefficients that take contrasting values of order one and of order $${\varepsilon^2}$$ε2 in different parts of the period cell. We provide a construction for the leading order term of the “operator asymptotics” of $${(\mathcal{A}^\varepsilon+I)^{-1}}$$(Aε+I)-1 in the sense of operator-norm convergence and prove order $${O(\varepsilon)}$$O(ε) remainder estimates.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/12f8f7fb604f67c54fbfdf2459cbb9a362e11805