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

Ali Sili

doi:10.5802/crmath.263

Partial differential equations

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

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.

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1-23.

Abstract
Résumé

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.

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 $3 d - 1 d$ intervenant localement sont identiques, à une condition aux limites près, la condition de Neumann homogène sur le bord de $C$ dans le problème $3 d - 1 d$ devant être remplacée dans le problème homogénéisé par une condition de périodicité.

Received: 2021-01-25
Revised: 2021-08-23
Accepted: 2021-08-23
Published online: 2022-01-26

DOI: 10.5802/crmath.263

Classification: 35B25, 35B27, 35B40, 35B45, 35J25, 35J57, 35J70, 35P20

Author's affiliations:

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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     title = {On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1--23},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2022},
     doi = {10.5802/crmath.263},
     language = {en},
}

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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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