{
  "id": "2006.04582",
  "version": "v1",
  "published": "2020-06-08T13:30:25.000Z",
  "updated": "2020-06-08T13:30:25.000Z",
  "title": "Exponential bounds for gradient of solutions to linear elliptic and parabolic equations",
  "authors": [
    "Kévin Le Balc'h"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we prove global gradient estimates for solutions to linear elliptic and parabolic equations. For a sufficiently smooth bounded convex domain $\\Omega \\subset \\mathbb{R}^N$, we show that a solution $\\phi \\in W_0^{1,\\infty}(\\Omega)$ to an appropriate elliptic equation $\\mathcal{L} \\phi = F$, with $F \\in L^{\\infty}(\\Omega;\\mathbb{R})$, satisfies $|\\nabla \\phi|_{\\infty} \\leq C |F|_{\\infty}$, with a positive constant $C = \\exp(C(\\mathcal{L})\\text{diam}(\\Omega))$. We also obtain similiar estimates in the parabolic setting. The proof of these exponential bounds relies on global gradient estimates inspired by a series of papers by Ben Andrews and Julie Clutterbuck. This work is motivated by a dual version of the Landis conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-08T13:30:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear elliptic",
      "parabolic equations",
      "global gradient estimates",
      "sufficiently smooth bounded convex domain",
      "appropriate elliptic equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}