{
  "id": "1704.04492",
  "version": "v1",
  "published": "2017-04-14T17:47:20.000Z",
  "updated": "2017-04-14T17:47:20.000Z",
  "title": "Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian",
  "authors": [
    "Hussien Abugirda",
    "Birzhan Ayanbayev",
    "Nikos Katzourakis"
  ],
  "comment": "15 pages, 2 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we consider the PDE system of vanishing normal projection of the Laplacian for $C^2$ maps $u : \\mathbb{R}^n \\supseteq \\Omega \\longrightarrow \\mathbb{R}^N$: \\[ [\\![\\mathrm{D} u]\\!]^\\bot \\Delta u = 0 \\ \\, \\text{ in }\\Omega. \\] This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the $p$-Laplace system for all $p\\in [2,\\infty]$. For $p=\\infty$, the $\\infty$-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial Calculus of Variations in $L^\\infty$. Herein we show that the image of a solution $u$ is piecewise affine if either the rank of $\\mathrm{D} u$ is equal to one or $n=2$ and $u$ has additively separated form. As a consequence we obtain corresponding flatness results for $p$-Harmonic maps, $p\\in [2,\\infty]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-14T17:47:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tangential laplacian",
      "laplace system",
      "vector field tangential",
      "vectorial calculus",
      "vanishing normal projection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}