{
  "id": "1012.0867",
  "version": "v2",
  "published": "2010-12-04T00:22:37.000Z",
  "updated": "2010-12-08T11:29:17.000Z",
  "title": "Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates",
  "authors": [
    "Xavier Cabre",
    "Yannick Sire"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This is the first of two articles dealing with the equation $(-\\Delta)^{s} v= f(v)$ in $\\mathbb{R}^{n}$, with $s\\in (0,1)$, where $(-\\Delta)^{s}$ stands for the fractional Laplacian ---the infinitesimal generator of a L\\'evy process. This equation can be realized as a local linear degenerate elliptic equation in $\\mathbb{R}^{n+1}_+$ together with a nonlinear Neumann boundary condition on $\\partial \\mathbb{R}^{n+1}_+=\\mathbb{R}^{n}$. In this first article, we establish necessary conditions on the nonlinearity $f$ to admit certain type of solutions, with special interest in bounded increasing solutions in all of $\\mathbb{R}$. These necessary conditions (which will be proven in a follow-up paper to be also sufficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian ---in the spirit of a result of Modica for the Laplacian. In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-12-08T11:29:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional laplacian",
      "maximum principles",
      "nonlinear equations",
      "hamiltonian estimates",
      "local linear degenerate elliptic equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.0867C"
    }
  }
}