{
  "id": "1303.0240",
  "version": "v2",
  "published": "2013-03-01T18:09:33.000Z",
  "updated": "2013-04-11T16:39:34.000Z",
  "title": "The Subelliptic $\\infty$-Laplace System on Carnot-Carathéodory Spaces",
  "authors": [
    "Nicholas Katzourakis"
  ],
  "comment": "16 pages, 2 figures, to appear in Advances in Nonlinear Analysis",
  "categories": [
    "math.AP"
  ],
  "abstract": "Given a Carnot-Carath\\'eodory space $\\Om \\sub \\R^n$ with associated vector fields $X=\\{X_1,...,X_m\\}$, we derive the subelliptic $\\infty$-Laplace system for mappings $u : \\Om \\larrow \\R^N$, which reads \\[ \\label{1} \\De^X_\\infty u \\, :=\\, \\Big(Xu \\ot Xu + \\|Xu\\|^2 [Xu]^\\bot \\ot I \\Big) : XX u\\, = \\, 0 \\tag{1} \\] in the limit of the subelliptic $p$-Laplacian as $p\\ri \\infty$. Here $Xu$ is the horizontal gradient and $[Xu]^\\bot$ is the projection on its nullspace. Next, we identify the Variational Principle characterizing \\eqref{1}, which is the \"Euler-Lagrange PDE\" of the supremal functional \\[ \\label{2} E_\\infty(u,\\Om)\\ := \\ \\|Xu\\|_{L^\\infty(\\Om)} \\tag{2} \\] for an appropriately defined notion of \\emph{Horizontally $\\infty$-Minimal Mappings}. We also establish a maximum principle for $\\|Xu\\|$ for solutions to \\eqref{1}. These results extend previous work of the author \\cite{K1, K2} on vector-valued Calculus of Variations in $L^\\infty$ from the Euclidean to the subelliptic setting.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-04-11T16:39:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J47",
      "35J62",
      "53C24",
      "49J99"
    ],
    "keywords": [
      "laplace system",
      "carnot-carathéodory spaces",
      "subelliptic",
      "carnot-caratheodory space",
      "variational principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.0240K"
    }
  }
}