Spectral analysis of one-dimensional high-contrast elliptic problems with periodic coefficients

TL;DR

This paper investigates the spectral properties of one-dimensional high-contrast periodic elliptic operators, revealing new effects due to non-uniform ellipticity and the existence of localized defect modes in the spectrum.

Contribution

It provides a detailed spectral analysis of high-contrast periodic elliptic operators in one dimension, highlighting differences from multi-dimensional models and identifying defect modes.

Findings

High-contrast coefficients lead to unique spectral effects.

Existence of localized eigenfunctions in spectral gaps.

Numerical evidence supports the theoretical predictions.

Abstract

We study the behaviour 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 non-uniform ellipticity of the family. The effective behaviour of such media in the vanishing period limit also differs notably from that of multi-dimensional 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 localised eigenfunctions ("defect modes"), whose eigenvalues situated in the gaps of the limit spectrum for the unperturbed problem.