{
  "id": "2410.12637",
  "version": "v1",
  "published": "2024-10-16T14:55:22.000Z",
  "updated": "2024-10-16T14:55:22.000Z",
  "title": "On solutions to a class of degenerate equations with the Grushin operator",
  "authors": [
    "Laura Abatangelo",
    "Alberto Ferrero",
    "Paolo Luzzini"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The Grushin Laplacian $- \\Delta_\\alpha $ is a degenerate elliptic operator in $\\mathbb{R}^{h+k}$ that degenerates on $\\{0\\} \\times \\mathbb{R}^k$. We consider weak solutions of $- \\Delta_\\alpha u= Vu$ in an open bounded connected domain $\\Omega$ with $V \\in W^{1,\\sigma}(\\Omega)$ and $\\sigma > Q/2$, where $Q = h + (1+\\alpha)k$ is the so-called homogeneous dimension of $\\mathbb{R}^{h+k}$. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of $\\Omega$. As an application we derive strong unique continuation properties for solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-16T14:55:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35H10",
      "35J70",
      "35B40",
      "35A16"
    ],
    "keywords": [
      "degenerate equations",
      "grushin operator",
      "derive strong unique continuation properties",
      "almgren-type monotonicity formula",
      "degenerate elliptic operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}