{
  "id": "2301.04226",
  "version": "v1",
  "published": "2023-01-10T22:11:41.000Z",
  "updated": "2023-01-10T22:11:41.000Z",
  "title": "On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems",
  "authors": [
    "Kaïs Ammari",
    "Ali Sili"
  ],
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "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 $ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-10T22:11:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B25",
      "35B27",
      "35B40",
      "35B45",
      "35J25",
      "35J57",
      "35J70",
      "35P20"
    ],
    "keywords": [
      "singular perturbation problems",
      "degenerate operator",
      "limit spectrum",
      "periodic homogenization",
      "stationary heat equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}