{
  "id": "2109.01455",
  "version": "v1",
  "published": "2021-09-03T11:35:03.000Z",
  "updated": "2021-09-03T11:35:03.000Z",
  "title": "Existence of an optimal domain for minimizing the fundamental tone of a clamped plate of prescribed volume in arbitrary dimension",
  "authors": [
    "Kathrin Stollenwerk"
  ],
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "In the 19th century, Lord Rayleigh conjectured that among all clamped plates with given area, the disk minimizes the fundamental tone. In the 1990s, N. S. Nadirashvili proved the conjecture in $\\mathbb{R}^2$ and M. S. Ashbaugh und R. D. Benguria gave a proof in $\\mathbb{R}^2$ and $\\mathbb{R}^3$. In the present paper, we prove existence of an optimal domain for minimizing the fundamental tone among all open and bounded subsets of $\\mathbb{R}^n$, $n\\geq 4$, with given measure. We formulate the minimization of the fundamental tone of a clamped plate as a free boundary value problem with a penalization term for the volume constraint. As the penalization parameter becomes small we show that the optimal shape problem is solved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-03T11:35:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fundamental tone",
      "clamped plate",
      "optimal domain",
      "arbitrary dimension",
      "prescribed volume"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}