{
  "id": "1604.04385",
  "version": "v1",
  "published": "2016-04-15T07:37:13.000Z",
  "updated": "2016-04-15T07:37:13.000Z",
  "title": "D-Solutions to the System of Vectorial Calculus of Variations in L-infinity via the Baire Category Method for the Singular Values",
  "authors": [
    "Gisella Croce",
    "Nikos Katzourakis",
    "Giovanni Pisante"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "For $\\mathrm{H}\\in C^2(\\R^{N\\by n})$ and $u :\\Om \\sub \\R^n \\larrow \\R^N$, consider the system \\[ \\label{1}\\mathrm{A}\\_\\infty u\\, :=\\,\\Big(\\mathrm{H}\\_P \\ot \\mathrm{H}\\_P + \\mathrm{H}[\\mathrm{H}\\_P]^\\bot \\mathrm{H}\\_{PP}\\Big)(\\D u):\\D^2u\\, =\\,0. \\tag{1}\\]The PDE system \\eqref{1} is associated to the supremal functional\\[ \\label{2}\\ \\ \\ \\ \\mathrm{E}\\_\\infty(u,\\Omega')\\, =\\, \\big\\| \\mathrm{H}(\\D u)\\|\\_{L^\\infty(\\Om')}, \\ \\ \\ u\\in W^{1,\\infty}\\_{\\text{loc}}(\\Om,\\R^N),\\ \\Omega'\\Subset \\Omega, \\tag{2} \\]and first arose in recent work of the 2nd author as the analogue of the Euler-Lagrange equation. Herein we employ the Dacorogna-Marcellini Baire Category method to construct $\\mD$-solutions to the Dirichlet problem for \\eqref{1}, an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our $\\mD$-solutions to \\eqref{1} are $W^{1,\\infty}$-submersions corresponding to \"critical points\" of \\eqref{2} and are obtained without any convexity hypotheses. Along the way we establish a result of independent interest by proving existence of strong solutions to the singular value problem for general dimensions $n\\neq N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-15T07:37:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vectorial calculus",
      "d-solutions",
      "variations",
      "l-infinity",
      "dacorogna-marcellini baire category method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160404385C"
    }
  }
}