{
  "id": "2109.11331",
  "version": "v1",
  "published": "2021-09-23T12:22:48.000Z",
  "updated": "2021-09-23T12:22:48.000Z",
  "title": "Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries",
  "authors": [
    "Martino Bardi",
    "Alessandro Goffi"
  ],
  "comment": "29 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "The paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, that under a suitable bound at infinity from above and, respectively, from below, they must be constant. In a previous paper we proved an abstract result and discussed operators on the Heisenberg group. Here we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and zero-th order terms in the equations and discuss their optimality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-23T12:22:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "carnot group",
      "hörmander vector fields",
      "fully nonlinear equations",
      "grushin geometries",
      "liouville results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}