{
  "id": "1207.5985",
  "version": "v1",
  "published": "2012-07-25T13:24:42.000Z",
  "updated": "2012-07-25T13:24:42.000Z",
  "title": "The Dirichlet problem for the fractional Laplacian: regularity up to the boundary",
  "authors": [
    "Xavier Ros-Oton",
    "Joaquim Serra"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if $u$ is a solution of $(-\\Delta)^s u = g$ in $\\Omega$, $u \\equiv 0$ in $\\R^n\\setminus\\Omega$, for some $s\\in(0,1)$ and $g \\in L^\\infty(\\Omega)$, then $u$ is $C^s(\\R^n)$ and $u/\\delta^s|_{\\Omega}$ is $C^\\alpha$ up to the boundary $\\partial\\Omega$ for some $\\alpha\\in(0,1)$, where $\\delta(x)={\\rm dist}(x,\\partial\\Omega)$. For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreover, under further regularity assumptions on $g$ we obtain higher order H\\\"older estimates for $u$ and $u/\\delta^s$. Namely, the $C^\\beta$ norms of $u$ and $u/\\delta^s$ in the sets $\\{x\\in\\Omega : \\delta(x)\\geq\\rho\\}$ are controlled by $C\\rho^{s-\\beta}$ and $C\\rho^{\\alpha-\\beta}$, respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian \\cite{RS-CRAS,RS}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-07-25T13:24:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional laplacian",
      "dirichlet problem",
      "krylov boundary harnack method",
      "crucial tools",
      "regularity results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.5985R"
    }
  }
}