{
  "id": "1401.4521",
  "version": "v3",
  "published": "2014-01-18T08:03:30.000Z",
  "updated": "2014-04-16T08:01:21.000Z",
  "title": "Regularity for fully nonlinear nonlocal parabolic equations with rough kernels",
  "authors": [
    "Joaquim Serra"
  ],
  "comment": "Some typos fixed and proof of Proposition 4.5 simplified",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $u_t = \\I u$, where $\\I$ is translation invariant and elliptic with respect to the class $\\mathcal L_0(\\sigma)$ of Caffarelli and Silvestre, $\\sigma\\in(0,2)$ being the order of $\\I$. We prove that if $u$ is a viscosity solution in $B_1 \\times (-1,0]$ which is merely bounded in $\\R^n \\times (-1,0]$, then $u$ is $C^\\beta$ in space and $C^{\\beta/\\sigma}$ in time in $\\overline{B_{1/2}} \\times [-1/2,0]$, for all $\\beta< \\min\\{\\sigma, 1+\\alpha\\}$, where $\\alpha>0$. Our proof combines a Liouville type theorem ---relaying on the nonlocal parabolic $C^\\alpha$ estimate of Chang and D\\'avila--- and a blow up and compactness argument.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-04-16T08:01:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fully nonlinear nonlocal parabolic equations",
      "rough kernels",
      "regularity",
      "fully nonlinear parabolic integro-differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.4521S"
    }
  }
}