On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems

Abstract

In this paper we perform the analysis of the spectrum of a degenerate operator $A_\var$ corresponding to the stationary heat equation in a $\var$-periodic composite medium having two components with high contrast diffusivity. We prove that although $ A_\var$ is a bounded self-adjoint operator with compact resolvent, the limits of its eigenvalues when the size $\var$ of the medium tends to zero, make up a part of the spectrum of a unbounded operator $ A_0$, namely the eigenvalues of $ A_0$ located on the left of the first eigenvalue of the bi-dimensional Laplacian with homogeneous Dirichlet condition on the boundary of the representative cell. We also show that the homogenized problem does not differ in any way from the one-dimensional problem obtained in the study of the local reduction of dimension induced by the homogenization.