{
  "id": "2002.06472",
  "version": "v1",
  "published": "2020-02-15T23:44:40.000Z",
  "updated": "2020-02-15T23:44:40.000Z",
  "title": "First Robin Eigenvalue of the $p$-Laplacian on Riemannian Manifolds",
  "authors": [
    "Xiaolong Li",
    "Kui Wang"
  ],
  "comment": "comments are welcome",
  "categories": [
    "math.AP",
    "math.DG",
    "math.SP"
  ],
  "abstract": "We consider the first Robin eigenvalue $\\lambda_p(M,\\alpha)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty boundary, with $\\alpha \\in \\mathbb{R}$ being the Robin parameter. We prove eigenvalue comparison theorems of Cheng type for $\\lambda_p(M,\\alpha)$. For $\\alpha>0$, we establish sharp lower bound estimates of $\\lambda_p(M,\\alpha)$ in terms of the dimension, inradius, Ricci lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. For $\\alpha<0$, the lower bound becomes an upper bound. Our results cover corresponding comparison theorems for the first Dirichlet eigenvalue of the $p$-Laplacian when letting $\\alpha \\to +\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-15T23:44:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "35P30",
      "58C40",
      "58J50"
    ],
    "keywords": [
      "first robin eigenvalue",
      "riemannian manifold",
      "sharp lower bound estimates",
      "cover corresponding comparison theorems",
      "boundary mean curvature lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}