{
  "id": "2305.07201",
  "version": "v1",
  "published": "2023-05-12T02:24:16.000Z",
  "updated": "2023-05-12T02:24:16.000Z",
  "title": "The obstacle problem for a higher order fractional Laplacian",
  "authors": [
    "Donatella Danielli",
    "Alaa Haj Ali",
    "Arshak Petrosyan"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the obstacle problem for the fractional Laplace operator $(-\\Delta)^s$ in the Euclidian space $\\mathbb{R}^n$ in the case where $1<s<2$. As first observed in \\cite{Y}, the problem can be extended to the upper half-space $\\mathbb{R}_+^{n+1}$ to obtain a thin obstacle problem for the weighted biLaplace operator $\\Delta^2_b U$, where $\\Delta_b U=y^{-b}\\nabla \\cdot (y^b \\nabla U)$. Such a problem arises in connection with unilateral phenomena for elastic, homogenous, and isotropic flat plates. We establish the well-posedness and $C_{loc}^{1,1}(\\R^n) \\cap H^{1+s}(\\R^n)$-regularity of the solution. By writing the solutions in terms of Riesz potentials of suitable local measures, we can base our proofs on tools from potential theory, such as a continuity principle and a maximum principle. Finally, we deduce the regularity of the extension problem to the higher dimensional upper half space. This gives an extension of Schild's work in \\cite{Sc1} and \\cite{Sc2} from the case $b=0$ to the general case $-1<b<1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-12T02:24:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B65",
      "35R35",
      "35J35",
      "35R11"
    ],
    "keywords": [
      "higher order fractional laplacian",
      "higher dimensional upper half space",
      "fractional laplace operator",
      "thin obstacle problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}