{
  "id": "2106.10216",
  "version": "v1",
  "published": "2021-06-18T16:40:45.000Z",
  "updated": "2021-06-18T16:40:45.000Z",
  "title": "Operator estimates for homogenization of the Robin Laplacian in a perforated domain",
  "authors": [
    "Andrii Khrabustovskyi",
    "Michael Plum"
  ],
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "Let $\\varepsilon>0$ be a small parameter. We consider the domain $\\Omega_\\varepsilon:=\\Omega\\setminus D_\\varepsilon$, where $\\Omega$ is an open domain in $\\mathbb{R}^n$, and $D_\\varepsilon$ is a family of small balls of the radius $d_\\varepsilon=o(\\varepsilon)$ distributed periodically with period $\\varepsilon$. Let $\\Delta_\\varepsilon$ be the Laplace operator in $\\Omega_\\varepsilon$ subject to the Robin condition ${\\partial u\\over \\partial n}+\\gamma_\\varepsilon u = 0$ with $\\gamma_\\varepsilon\\ge 0$ on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have shown that, under appropriate assumptions on $d_\\varepsilon$ and $\\gamma_\\varepsilon$, the operator $\\Delta_\\varepsilon$ converges in the strong resolvent sense to the sum of the Dirichlet Laplacian in $\\Omega$ and a constant potential. We improve this result deriving estimates on the rate of convergence in terms of $L^2\\to L^2$ and $L^2\\to H^1$ operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-18T16:40:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B27",
      "35B40",
      "35P05",
      "47A55"
    ],
    "keywords": [
      "operator estimates",
      "robin laplacian",
      "perforated domain",
      "homogenization",
      "strong resolvent sense"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}