{
  "id": "2307.13641",
  "version": "v1",
  "published": "2023-07-25T16:45:58.000Z",
  "updated": "2023-07-25T16:45:58.000Z",
  "title": "On the Poincaré inequality on open sets in $\\mathbb{R}^n$",
  "authors": [
    "A. -K. Gallagher"
  ],
  "categories": [
    "math.AP",
    "math.CV",
    "math.SP"
  ],
  "abstract": "We show that the Poincar\\'{e} inequality holds on an open set $D\\subset\\mathbb{R}^n$ if and only if $D$ admits a smooth, bounded function whose Laplacian has a positive lower bound on $D$. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on $D$ is equivalent to the finiteness of the strict inradius of $D$ measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet--Laplacian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-25T16:45:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "31B99",
      "32W05"
    ],
    "keywords": [
      "open set",
      "sharp upper bound",
      "inequality holds",
      "smallest eigenvalue",
      "newtonian capacity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}