{
  "id": "2103.15911",
  "version": "v1",
  "published": "2021-03-29T19:40:27.000Z",
  "updated": "2021-03-29T19:40:27.000Z",
  "title": "Generalised vectorial $\\infty$-eigenvalue nonlinear problems for $L^\\infty$ functionals",
  "authors": [
    "Nikos Katzourakis"
  ],
  "comment": "30 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega \\Subset \\mathbb R^n$, $f \\in C^1(\\mathbb R^{N\\times n})$ and $g\\in C^1(\\mathbb R^N)$, where $N,n \\in \\mathbb N$. We study the minimisation problem of finding $u \\in W^{1,\\infty}_0(\\Omega;\\mathbb R^N)$ that satisfies \\[ \\big\\| f(\\mathrm D u) \\big\\|_{L^\\infty(\\Omega)} \\! = \\inf \\Big\\{\\big\\| f(\\mathrm D v) \\big\\|_{L^\\infty(\\Omega)} \\! : \\ v \\! \\in W^{1,\\infty}_0(\\Omega;\\mathbb R^N), \\, \\| g(v) \\|_{L^\\infty(\\Omega)}\\! =1\\Big\\}, \\] under natural assumptions on $f,g$. This includes the $\\infty$-eigenvalue problem as a special case. Herein we prove existence of a minimiser $u_\\infty$ with extra properties, derived as the limit of minimisers of approximating constrained $L^p$ problems as $p\\to \\infty$. A central contribution and novelty of this work is that $u_\\infty$ is shown to solve a divergence PDE with measure coefficients, whose leading term is a divergence counterpart equation of the non-divergence $\\infty$-Laplacian. Our results are new even in the scalar case of the $\\infty$-eigenvalue problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-29T19:40:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D30",
      "35D40",
      "35J47",
      "35J92",
      "35J70",
      "35J99",
      "35P30"
    ],
    "keywords": [
      "eigenvalue nonlinear problems",
      "generalised vectorial",
      "functionals",
      "eigenvalue problem",
      "divergence counterpart equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}