{
  "id": "1701.03348",
  "version": "v1",
  "published": "2017-01-12T14:27:04.000Z",
  "updated": "2017-01-12T14:27:04.000Z",
  "title": "Existence, Uniqueness and Structure of Second Order absolute minimisers",
  "authors": [
    "Nikos Katzourakis",
    "Roger Moser"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega \\subseteq \\mathbb{R}^n$ be a bounded open $C^{1,1}$ set. In this paper we prove the existence of a unique second order absolute minimiser $u_\\infty$ of the functional \\[ \\mathrm{E}_\\infty (u,\\mathcal{O})\\, :=\\, \\| \\mathrm{F}(\\cdot, \\Delta u) \\|_{L^\\infty( \\mathcal{O} )}, \\ \\ \\ \\mathcal{O} \\subseteq \\Omega \\text{ measurable}, \\] with prescribed boundary conditions for $u$ and $\\mathrm{D} u$ on $\\partial \\Omega$ and under natural assumptions on $\\mathrm{F}$. We also show that $u_\\infty$ is partially smooth and there exists a harmonic function $f_\\infty \\in L^1(\\Omega)$ such that \\[ \\mathrm{F}(x, \\Delta u_\\infty(x)) \\, =\\, e_\\infty\\, \\mathrm{sgn}\\big(f_\\infty(x)\\big) \\] for all $x \\in \\{f_\\infty \\neq 0\\}$, where $e_\\infty$ is the infimum of the global energy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-12T14:27:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unique second order absolute minimiser",
      "uniqueness",
      "prescribed boundary conditions",
      "natural assumptions",
      "harmonic function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}