Spectral Convergence for High‐Contrast Elliptic Periodic Problems with a Defect Via Homogenization

Abstract

We consider an eigenvalue problem for a divergence-form elliptic operator Aε that has high-contrast periodic coefficients with period ε in each coordinate, where ε is a small parameter. The coefficients are perturbed on a bounded domain of “order one” size. The local perturbation of coefficients for such an operator could result in the emergence of localized waves—eigenfunctions whose corresponding eigenvalues lie in the gaps of the Floquet–Bloch spectrum. For the so-called double porosity-type scaling, we prove that the eigenfunctions decay exponentially at infinity, uniformly in ε Then, using the tools of twoscale convergence for high-contrast homogenization, we prove the strong two-scale compactness of the eigenfunctions of Aε. This implies that the eigenfunctions converge in the sense of strong two-scale convergence to the eigenfunctions of a two-scale limit homogenized operator A0, consequently establishing “asymptotic one-to-one correspondence” between the eigenvalues and the eigenfunctions of the operators Aε and A0. We also prove, by direct means, the stability of the essential spectrum of the homogenized operator with respect to local perturbation of its coefficients. This allows us to establish not only the strong two-scale resolvent convergence of Aε to A0 but also the Hausdorff convergence of the spectra of Aε to the spectrum of A0, preserving the multiplicity of the isolated eigenvalues.