On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems
Ali Sili · Comptes Rendus Mathématique · 2022
In this paper we perform the analysis of the spectrum of a degenerate operator A ε corresponding to the stationary heat equation in a ε -periodic composite medium having two components with high contrast diffusivity. We prove that although A ε is a self-adjoint operator with compact resolvent, its limit A 0 when the size ε of the medium tends to zero is a non self-adjoint operator whose spectrum is bounded by positive constants depending on the first eigenvalue of the one-dimensional Laplacian in H 0 1 ( 0 , L ) and the first eigenvalue of the bi-dimensional Laplacian with mixed boundary conditions on the representative cell C . Furthermore, we show that the homogenized problem and the one-dimensional limit problem obtained by the reduction of dimension 3 d - 1 d occurring locally are identical except for one boundary condition which is a homogeneous Neumann condition on the boundary of C in the 3 d - 1 d problem and a periodicity condition in the case of homogenization.