Comptes Rendus
Partial differential equations
On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1-23.

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 3d-1d occurring locally are identical except for one boundary condition which is a homogeneous Neumann condition on the boundary of C in the 3d-1d problem and a periodicity condition in the case of homogenization.

Dans ce travail, nous analysons le spectre d’un opérateur dégénéré A ε correspondant à l’équation de la chaleur stationnaire dans un milieu composite ε-périodique ayant deux composantes avec des coefficients de conductivité à fort contraste. Nous montrons que bien que 𝒜 ε soit un opérateur auto-adjoint à résolvante compacte, sa limite A 0 lorsque la période ε tend vers 0 est un opérateur non auto-adjoint dont le spectre est borné par des constantes positives ne dépendant que de la première valeur propre du Laplacien uni-dimensionnel dans H 0 1 (0,L) et de la première valeur propre du Laplacien bi-dimensionnel avec conditions au bord mixtes sur la cellule de référence C. Nous montrons en outre que le problème homogénéisé et le problème limite obtenu après réduction de dimension 3d-1d intervenant localement sont identiques, à une condition aux limites près, la condition de Neumann homogène sur le bord de C dans le problème 3d-1d devant être remplacée dans le problème homogénéisé par une condition de périodicité.

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DOI: 10.5802/crmath.263
Classification: 35B25, 35B27, 35B40, 35B45, 35J25, 35J57, 35J70, 35P20
Ali Sili 1

1 Institut de Mathématiques de Marseille (I2M), UMR 7373, Aix-Marseille Université, CNRS, CMI, 39 rue F. Joliot-Curie, 13453 Marseille cedex 13, France.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Ali Sili. On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1-23. doi : 10.5802/crmath.263. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.263/

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