{
  "id": "2011.14491",
  "version": "v1",
  "published": "2020-11-30T01:26:58.000Z",
  "updated": "2020-11-30T01:26:58.000Z",
  "title": "Bounded weak solutions to elliptic PDE with data in Orlicz spaces",
  "authors": [
    "David Cruz-Uribe",
    "Scott Rodney"
  ],
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "A classical regularity result is that non-negative solutions to the Dirichlet problem $\\Delta u =f$ in a bounded domain $\\Omega$, where $f\\in L^q(\\Omega)$, $q>\\frac{n}2$, satisfy $\\|u\\|_{L^\\infty(\\Omega)} \\leq C\\|f\\|_{L^q(\\Omega)}$. We extend this result in three ways: we replace the Laplacian with a degenerate elliptic operator; we show that we can take the data $f$ in an Orlicz space $L^A(\\Omega)$ that lies strictly between $L^{\\frac{n}{2}}(\\Omega)$ and $L^q(\\Omega)$, $q>\\frac{n}2$; and we show that that we can replace the $L^A$ norm in the right-hand side by a smaller expression involving the logarithm of the \"entropy bump\" $\\|f\\|_{L^A(\\Omega)}/\\|f\\|_{L^{\\frac{n}{2}}(\\Omega)}$, generalizing a result due to Xu.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-30T01:26:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B45",
      "35D30",
      "35J25",
      "46E30"
    ],
    "keywords": [
      "bounded weak solutions",
      "orlicz space",
      "elliptic pde",
      "degenerate elliptic operator",
      "dirichlet problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}