{
  "id": "1910.04905",
  "version": "v1",
  "published": "2019-10-10T23:15:03.000Z",
  "updated": "2019-10-10T23:15:03.000Z",
  "title": "Localization of the first eigenfunction of a convex domain",
  "authors": [
    "Thomas Beck"
  ],
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "We study the first Dirichlet eigenfunction of the Laplacian in a $n$-dimensional convex domain. For domains of a fixed inner radius, estimates of Chiti imply that the ratio of the $L^2$-norm and $L^{\\infty}$-norm of the eigenfunction is minimized when the domain is a ball. However, when the eccentricity of the domain is large the eigenfunction should spread out at a certain scale and this ratio should increase. We make this precise by obtaining a lower bound on the $L^2$-norm of the eigenfunction and show that the eigenfunction cannot localize to too small a subset of the domain. As a consequence, we settle a conjecture of van den Berg, in the general $n$-dimensional case. The main feature of the proof is to obtain sufficiently sharp estimates on the first eigenvalue in order to estimate the first derivatives of the eigenfunction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-10T23:15:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first eigenfunction",
      "localization",
      "first dirichlet eigenfunction",
      "van den berg",
      "dimensional convex domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}