Asymptotic analysis of some spectral problems in high contrast homogenisation and in thin domains

We study the spectral properties of two problems involving small parameters. The first one is an eigenvalue problem for a divergence form elliptic operator Aε with high contrast periodic coefficients of 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 in the gaps of the Floquet-Bloch spectrum. We prove that, for the so-called double porosity type scaling, the eigenfunctions decay exponentially at in infinity, uniformly in ε. Then, using the tools of twoscale convergence for high contrast homogenisation, we prove the strong twoscale convergence of the eigenfunctions of Aε 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 Hausdor convergence of the spectra of Aε to the spectrum of A₀ , preserving the multiplicity of the isolated eigenvalues. As the second problem we consider the eigenvalue problem for the Laplacian in a network of thin domains with Dirichlet boundary conditions. We construct an asymptotic solution to the problem using the method of matched asymptotic expansions to obtain appropriate boundary conditions for the terms of the asymptotics near the junctions of thin domains. We justify the asymptotics and prove the error bound of order h3=2 , where h is the width of thin domains.