Localized Modes Due to Defects in High Contrast Periodic Media Via Two-Scale Homogenization

Abstract

The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. For a high contrast small e-size periodicity and a finite size defect we consider the critical e2-scaling for the contrast. We employ two-scale homogenization for deriving asymptotically explicit limit equations for the localized modes and associated eigenvalues. Those are expressed in terms of the eigenvalues and eigenfunctions of a perturbed version of a two-scale limit operator introduced by V. V. Zhikov with an emergent explicit nonlinear dependence on the spectral parameter for the spectral problem at the macroscale. Using the method of asymptotic expansions supplemented by a high contrast boundary layer analysis, we establish the existence of the actual eigenvalues near the eigenvalues of the limit operator, with “e square root” error bounds. An example for circular or spherical defects in a periodic medium with isotropic homogenized properties is given.