{
  "id": "1811.10024",
  "version": "v1",
  "published": "2018-11-25T14:43:22.000Z",
  "updated": "2018-11-25T14:43:22.000Z",
  "title": "On the first eigenvalue of the normalized p-Laplacian",
  "authors": [
    "Graziano Crasta",
    "Ilaria Fragalà",
    "Bernd Kawohl"
  ],
  "comment": "13 pages, 2 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove that, if $\\Omega$ is an open bounded domain with smooth and connected boundary, for every $p \\in (1, + \\infty)$ the first Dirichlet eigenvalue of the normalized $p$-Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (non-optimal) lower bound for the eigenvalue in terms of the measure of $\\Omega$, and we address the open problem of proving a Faber-Krahn type inequality with balls as optimal domains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-25T14:43:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49K20",
      "35J60",
      "47J10"
    ],
    "keywords": [
      "first eigenvalue",
      "normalized p-laplacian",
      "faber-krahn type inequality",
      "first dirichlet eigenvalue",
      "optimal domains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}