{
  "id": "1412.0275",
  "version": "v1",
  "published": "2014-11-30T20:12:21.000Z",
  "updated": "2014-11-30T20:12:21.000Z",
  "title": "Boundary regularity for the fractional heat equation",
  "authors": [
    "Xavier Fernández-Real",
    "Xavier Ros-Oton"
  ],
  "comment": "This work is part of the bachelor's degree thesis of the first author",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the regularity up to the boundary of solutions to fractional heat equation in bounded $C^{1,1}$ domains. More precisely, we consider solutions to $\\partial_t u + (-\\Delta)^s u=0 \\textrm{ in }\\Omega,\\ t > 0$, with zero Dirichlet conditions in $\\mathbb{R}^n\\setminus \\Omega$ and with initial data $u_0\\in L^2(\\Omega)$. Using the results of the second author and Serra for the elliptic problem, we show that for all $t>0$ we have $u(\\cdot, t)\\in C^s(\\mathbb{R}^n)$ and $u(\\cdot, t)/\\delta^s \\in C^{s-\\epsilon}(\\overline\\Omega)$ for any $\\epsilon > 0$ and $\\delta(x) = \\textrm{dist}(x,\\partial\\Omega)$. Our regularity results apply not only to the fractional Laplacian but also to more general integro-differential operators, namely those corresponding to stable L\\'evy processes. As a consequence of our results, we show that solutions to the fractional heat equation satisfy a Pohozaev-type identity for positive times.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-30T20:12:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boundary regularity",
      "fractional heat equation satisfy",
      "zero dirichlet conditions",
      "general integro-differential operators",
      "stable levy processes"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.0275F"
    }
  }
}