{
  "id": "2209.08775",
  "version": "v1",
  "published": "2022-09-19T06:02:59.000Z",
  "updated": "2022-09-19T06:02:59.000Z",
  "title": "Operator estimates for Neumann sieve problem",
  "authors": [
    "Andrii Khrabustovskyi"
  ],
  "comment": "33 pages, 3 figures",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "Let $\\Omega$ be a domain in $\\mathbb{R}^n$, $\\Gamma$ be a hyperplane intersecting $\\Omega$, $\\varepsilon>0$ be a small parameter, and $D_{k,\\varepsilon}$, $k=1,2,3\\dots$ be a family of small \"holes\" in $\\Gamma\\cap\\Omega$; when $\\varepsilon \\to 0$, the number of holes tends to infinity, while their diameters tends to zero. Let $\\mathscr{A}_\\varepsilon$ be the Neumann Laplacian in the perforated domain $\\Omega_\\varepsilon=\\Omega\\setminus\\Gamma_\\varepsilon$, where $\\Gamma_\\varepsilon=\\Gamma\\setminus (\\cup_k D_{k,\\varepsilon})$ (\"sieve\"). It is well-known that if the sizes of holes are carefully chosen, $\\mathscr{A}_\\varepsilon$ converges in the strong resolvent sense to the Laplacian on $\\Omega\\setminus\\Gamma$ subject to the so-called $\\delta'$-conditions on $\\Gamma$. In the current work we improve this result: under rather general assumptions on the shapes and locations of the holes we derive estimates on the rate of convergence in terms of $L^2\\to L^2$ and $L^2\\to H^1$ operator norms; in the latter case a special corrector is required.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-19T06:02:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B27",
      "35B40",
      "35P05",
      "47A55"
    ],
    "keywords": [
      "neumann sieve problem",
      "operator estimates",
      "strong resolvent sense",
      "special corrector",
      "diameters tends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}