{
  "id": "1609.08186",
  "version": "v1",
  "published": "2016-09-26T20:48:21.000Z",
  "updated": "2016-09-26T20:48:21.000Z",
  "title": "Extremal functions for Morrey's inequality in convex domains",
  "authors": [
    "Ryan Hynd",
    "Erik Lindgren"
  ],
  "comment": "24 pages, 7 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "For a bounded domain $\\Omega\\subset \\mathbb{R}^n$ and $p>n$, Morrey's inequality implies that there is $c>0$ such that $$ c\\|u\\|^p_{\\infty}\\le \\int_\\Omega|Du|^pdx $$ for each $u$ belonging to the Sobolev space $W^{1,p}_0(\\Omega)$. We show that the ratio of any two extremal functions is constant provided that $\\Omega$ is convex. We also explain why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this property. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the $p$-Laplacian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-26T20:48:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extremal functions",
      "convex domains",
      "morreys inequality implies",
      "greens function",
      "optimization problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}