On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917672788164608 |
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| author | Ammari, Kaïs Sili, Ali |
| author_facet | Ammari, Kaïs Sili, Ali |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_04226 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems Ammari, Kaïs Sili, Ali Analysis of PDEs Spectral Theory 35B25, 35B27, 35B40, 35B45, 35J25, 35J57, 35J70, 35P20 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. |
| title | On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems |
| topic | Analysis of PDEs Spectral Theory 35B25, 35B27, 35B40, 35B45, 35J25, 35J57, 35J70, 35P20 |
| url | https://arxiv.org/abs/2301.04226 |