{
  "id": "1611.05936",
  "version": "v1",
  "published": "2016-11-18T00:45:24.000Z",
  "updated": "2016-11-18T00:45:24.000Z",
  "title": "A Pointwise Characterisation of the PDE System of Vectorial Calculus of Variations in $L^\\infty$",
  "authors": [
    "Birzhan Ayanbayev",
    "Nikos Katzourakis"
  ],
  "comment": "13 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $n,N\\in \\mathbb{N}$ with $\\Omega \\subseteq \\mathbb{R}^n$ open. Given $H \\in C^2(\\Omega \\times \\mathbb{R}^N\\times \\mathbb{R}^{Nn}),$ we consider the functional \\[ \\tag{1} \\label{1} E_\\infty (u,\\mathcal{O})\\, :=\\, \\underset{\\mathcal{O}}{\\mathrm{ess}\\,\\sup}\\, H (\\cdot,u,\\mathrm{D} u) ,\\ \\ \\ u\\in W^{1,\\infty}_\\text{loc}(\\Omega,\\mathbb{R}^N),\\ \\ \\ \\mathcal{O} \\Subset \\Omega. \\] The associated PDE system which plays the role of Euler-Lagrange equations in $L^\\infty$ is \\[ \\label{2} \\tag{2} \\left\\{ \\begin{array}{r} H_{P}(\\cdot, u, \\mathrm{D}u)\\, \\mathrm{D} \\big(H(\\cdot, u, \\mathrm{D} u)\\big) \\, = \\, 0, \\ \\ \\ H(\\cdot, u, \\mathrm{D} u) \\, [\\![H_{P}(\\cdot, u, \\mathrm{D} u)]\\!]^\\bot \\Big(\\mathrm{Div}\\big(H_{P}(\\cdot, u, \\mathrm{D} u)\\big)- H_{\\eta}(\\cdot, u, \\mathrm{D} u)\\Big)\\, =\\, 0, \\end{array} \\right. \\] where $[\\![A]\\!]^\\bot := \\mathrm{Proj}_{R(A)^\\bot}$. Herein we establish that generalised solutions to \\eqref{2} can be characterised as local minimisers of \\eqref{1} for appropriate classes of affine variations of the energy. Generalised solutions to \\eqref{2} are understood as $\\mathcal{D}$-solutions, a general framework recently introduced by one of the authors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-18T00:45:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vectorial calculus",
      "pointwise characterisation",
      "generalised solutions",
      "euler-lagrange equations",
      "appropriate classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}