Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients

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

Spectral Analysis of One-Dimensional High-Contrast Elliptic Problems with Periodic Coefficients

We study the behavior of the spectrum of a family of one-dimensional operators with periodic high-contrast coefficients as the period goes to zero, which may represent, e.g., the elastic or electromagnetic response of a two-component composite medium. Compared to the standard operators with moderate contrast, they exhibit a number of new effects due to the underlying nonuniform ellipticity of the family. The effective behavior of such media in the vanishing period limit also differs notably from that of multidimensional models investigated thus far by other authors, due to the fact that neither component of the composite forms a connected set. We then discuss a modified problem, where the equation coefficient is set to a positive constant on an interval that is independent of the period. Formal asymptotic analysis and numerical tests with finite elements suggest the existence of localized eigenfunctions (``defect modes''), whose eigenvalues are situated in the gaps of the limit spectrum for the unperturbed problem

On the existence of high-frequency boundary resonances in layered elastic media

We analyse the asymptotic behaviour of high-frequency vibrations of a three-dimensional layered elastic medium occupying the domain Ω=(−a,a)3, a>0. We show that in both cases of stress-free and zero-displacement boundary conditions on the boundary of Ω a version of the boundary spectrum, introduced in Allaire and Conca (1998 J. Math. Pures. Appl. 77, 153–208. (doi:10.1016/S0021-7824(98)80068-8)), is non-empty and part of it is located below the Bloch spectrum. For zero-displacement boundary conditions, this yields a new type of surface wave, which is absent in the case of a homogeneous medium