{
  "id": "1903.09526",
  "version": "v1",
  "published": "2019-03-22T14:23:19.000Z",
  "updated": "2019-03-22T14:23:19.000Z",
  "title": "Dirichlet-to-Neumann maps on Trees",
  "authors": [
    "Leandro M. Del Pezzo",
    "Nicolás Frevenza",
    "Julio D. Rossi"
  ],
  "comment": "27 pages. Keywords: Dirichlet-to-Neumann map, Mean value formulas, Equations on trees",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "In this paper we study the Dirichlet-to-Neumann map for solutions to mean value formulas on trees. We give two alternative definition of the Dirichlet-to-Neumann map. For the first definition (that involves the product of a \"gradient\" with a \"normal vector\") and for a linear mean value formula on the directed tree (taking into account only the successors of a given node) we obtain that the Dirichlet-to-Neumann map is given by $g\\mapsto cg'$ (here $c$ is an explicit constant). Notice that this is a local operator of order one. We also consider linear undirected mean value formulas (taking into account not only the successors but the ancestor and the successors of a given node) and prove a similar result. For this kind of mean value formula we include some existence and uniqueness results for the associated Dirichlet problem. Finally, we give an alternative definition of the Dirichlet-to-Neumann map (taking into account differences along a given branch of the tree). With this alternative definition, for a certain range of parameters, we obtain that the Dirichlet-to-Neumann map is given by a nonlocal operator (as happens for the classical Laplacian in the Euclidean space).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-22T14:23:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J05",
      "35R30",
      "31E05",
      "37E25"
    ],
    "keywords": [
      "dirichlet-to-neumann map",
      "alternative definition",
      "linear undirected mean value formulas",
      "linear mean value formula",
      "successors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}