{
  "id": "2107.12117",
  "version": "v1",
  "published": "2021-07-26T11:28:53.000Z",
  "updated": "2021-07-26T11:28:53.000Z",
  "title": "Eigenvalue Problems in $\\mathrm{L}^\\infty$: Optimality Conditions, Duality, and Relations with Optimal Transport",
  "authors": [
    "Leon Bungert",
    "Yury Korolev"
  ],
  "categories": [
    "math.AP",
    "math.OC",
    "math.SP"
  ],
  "abstract": "In this article we characterize the $\\mathrm{L}^\\infty$ eigenvalue problem associated to the Rayleigh quotient $\\left.{\\|\\nabla u\\|_{\\mathrm{L}^\\infty}}\\middle/{\\|u\\|_\\infty}\\right.$, defined on Lipschitz functions with homogeneous boundary conditions on a domain $\\Omega$. For this, we derive a novel fine characterization of the subdifferential of the functional $u\\mapsto\\|\\nabla u\\|_{\\mathrm{L}^\\infty}$. Using the concept of tangential gradients we show that it consists of Radon measures of the form $-\\operatorname{div}\\sigma$ where $\\sigma$ is a normalized divergence-measure field, in a suitable sense parallel to the gradient $\\nabla u$ and concentrated where it is maximal. We also investigate geometric relations between general minimizers of the Rayleigh quotient and suitable distance functions. Lastly, we investigate a \"dual\" Rayleigh quotient whose minimizers are subgradients of minimizers of the original quotient and solve an optimal transport problem associated to a generalized Kantorovich-Rubinstein norm. Our results apply to all minimizers of the Rayleigh quotient, including infinity ground states, infinity harmonic potentials, distance functions, etc., and generalize known results in the literature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-26T11:28:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A16",
      "35P30",
      "46N10",
      "47J10",
      "49R05"
    ],
    "keywords": [
      "eigenvalue problem",
      "optimality conditions",
      "rayleigh quotient",
      "minimizers",
      "distance functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}