{
  "id": "1801.09756",
  "version": "v1",
  "published": "2018-01-29T20:57:47.000Z",
  "updated": "2018-01-29T20:57:47.000Z",
  "title": "Counterexamples in Calculus of Variations in $L^\\infty$ through the vectorial Eikonal equation",
  "authors": [
    "Nikos Katzourakis",
    "Giles Shaw"
  ],
  "comment": "5 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We show that for any regular bounded domain $\\Omega\\subseteq \\mathbb R^n$, there exist infinitely many global diffeomorphisms equal to the identity on $\\partial \\Omega$ which solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the $\\infty$-Laplace system arising in vectorial Calculus of Variations in $L^\\infty$ does not suffice to characterise either limits of $p$-Harmonic maps as $p\\to \\infty$, or absolute minimisers in the sense of Aronsson.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-29T20:57:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vectorial eikonal equation",
      "variations",
      "counterexamples",
      "global diffeomorphisms equal",
      "regular bounded domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}