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

TL;DR

This paper investigates the spectral behavior of a degenerate operator in periodic homogenization, revealing how eigenvalues converge to those of an unbounded limit operator and showing the homogenized problem reduces to a one-dimensional case.

Contribution

It provides a detailed analysis of the limit spectrum of a degenerate operator in high-contrast periodic media, connecting it to the spectrum of an unbounded operator and simplifying the homogenized problem.

Findings

Eigenvalues of the operator converge to those of an unbounded limit operator.

The homogenized problem reduces to a one-dimensional problem.

The spectrum of the degenerate operator is characterized in the limit as the medium size tends to zero.

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.