{
  "id": "2006.06612",
  "version": "v1",
  "published": "2020-06-11T17:10:30.000Z",
  "updated": "2020-06-11T17:10:30.000Z",
  "title": "Liouville results for fully nonlinear equations modeled on Hörmander vector fields. I. The Heisenberg group",
  "authors": [
    "Martino Bardi",
    "Alessandro Goffi"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the H\\\"ormander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is done here for the vector fields that generate the Heisenberg group, giving explicit conditions on the sign and size of the first and zero-th order terms in the equation. The optimality of the conditions is shown via several examples. A sequel of this paper applies the methods to other Carnot groups and to Grushin geometries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-11T17:10:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hörmander vector fields",
      "fully nonlinear equations",
      "heisenberg group",
      "liouville results",
      "paper studies liouville properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}