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

2023/01/10 by Ammari, Kaïs, Sili, Ali
#35B25 #35B27 #35B40 #35B45 #35J25 #35J57 #35J70 #35P20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2301.04226

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 A₀, namely the eigenvalues of A₀ 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.

Related

• Homogenization of the eigenvalues of the Neumann-Poincaré operator
• Homogenization of nonlocal spectral problems
• Spectral convergence for high contrast elliptic periodic problems with a defect via homogenization
• Homogenization of spectral problems in bounded domains with doubly high contrasts
• Localised modes due to defects in high contrast periodic media via two-scale homogenization