Spectral convergence for high contrast elliptic periodic problems with a defect via homogenisation

We consider an eigenvalue problem for a divergence form elliptic operator Aε with 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 operator could result in emergence of localised waves eigenfunctions with corresponding eigenvalues lying in the gaps of the Floquet-Bloch spectrum. We prove that, for the so-called double porosity type scaling, the eigenfunctions decay exponentially at infinity, uniformly in ε. Then, using the tools of two-scale convergence for high contrast homogenisation, we prove the strong two-scale compactness of the eigenfunctions of Aε. This implies that the eigenfunctions converge in the sense of the strong two-scale convergence to the eigenfunctions of a two-scale limit homogenised operator A0, consequently establishing ‘asymptotic one-to-one correspondence’ between the eigenvalues and the eigenfunctions of these two operators. We also prove by direct means the stability of the essential spectrum of the homogenised operator with respect to the local perturbation of its coefficients. That 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.