{
  "id": "1910.02295",
  "version": "v1",
  "published": "2019-10-05T17:09:33.000Z",
  "updated": "2019-10-05T17:09:33.000Z",
  "title": "Sharp lower bound for the first eigenvalue of the Weighted $p$-Laplacian",
  "authors": [
    "Xiaolong Li",
    "Kui Wang"
  ],
  "comment": "Comments are welcome",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Lapacian operator with $1< p< \\infty$ on a compact Bakry-Emery manifold $(M^n,g,f)$ satisfying $\\Ric+\\nabla^2 f \\geq \\kappa \\, g$, provided that either $1<p \\leq 2$ or $\\kappa \\leq 0$. Same conclusions hold when the manifold has nonempty boundary if we assume it is strictly convex and put Neumann boundary conditions on it. For $1<p \\leq 2$, we provide a simple proof via the modulus of continuity estimates method. The proof for $\\kappa \\leq 0$ is based on a sharp gradient comparison theorem for the eigenfunction and a careful analysis of the underlying one-dimensional model equation. Our results generalize the work of Valtorta\\cite{Valtorta12} and Naber-Valtorta\\cite{NV14} for the $p$-Laplacian (namely $f=\\text{const}$), and the work of Bakry-Qian\\cite{BQ00} for the $f$-Laplacian (namely $p=2$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-05T17:09:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "35P30"
    ],
    "keywords": [
      "first eigenvalue",
      "sharp lower bound estimates",
      "sharp gradient comparison theorem",
      "one-dimensional model equation",
      "compact bakry-emery manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}