{
  "id": "2111.09232",
  "version": "v2",
  "published": "2021-11-17T16:45:13.000Z",
  "updated": "2022-01-27T16:06:47.000Z",
  "title": "Linearity of homogeneous solutions to degenerate elliptic equations in dimension three",
  "authors": [
    "Jose A. Galvez",
    "Pablo Mira"
  ],
  "comment": "21 pages, 7 figures. Minor wording changes with respect to the initial version",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Given a linear elliptic equation $\\sum a_{ij} u_{ij} =0$ in $\\mathbb{R}^3$, it is a classical problem to determine if its degree-one homogeneous solutions $u$ are linear. The answer is negative in general, by a construction of Martinez-Maure. In contrast, the answer is affirmative in the uniformly elliptic case, by a theorem of Han, Nadirashvili and Yuan, and it is a known open problem to determine the degenerate ellipticity condition on $(a_{ij})$ under which this theorem still holds. In this paper we solve this problem. We prove the linearity of $u$ under the following degenerate ellipticity condition for $(a_{ij})$, which is sharp by Martinez-Maure example: if $\\mathcal{K}$ denotes the ratio between the largest and smallest eigenvalues of $(a_{ij})$, we assume $\\mathcal{K}|_{\\mathcal{O}}$ lies in $L_{\\rm loc}^1$ for some connected open set $\\mathcal{O}\\subset \\mathbb{S}^2$ that intersects any configuration of four disjoint closed geodesic arcs of length $\\pi$ in $\\mathbb{S}^2$. Our results also give the sharpest possible version under which an old conjecture by Alexandrov, Koutroufiotis and Nirenberg (disproved by Martinez-Maure's example) holds.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-27T16:06:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J70",
      "35R05",
      "53A05",
      "53C42"
    ],
    "keywords": [
      "degenerate elliptic equations",
      "degenerate ellipticity condition",
      "linear elliptic equation",
      "disjoint closed geodesic arcs",
      "martinez-maure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}