{
  "id": "math/0202140",
  "version": "v2",
  "published": "2002-02-15T04:22:58.000Z",
  "updated": "2002-04-08T06:58:39.000Z",
  "title": "Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions",
  "authors": [
    "Andrew Hassell",
    "Terence Tao"
  ],
  "comment": "16 pages, 1 figure. Some minor errors and ambiguous notation corrected, and the diagram compressed",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $\\lambda$. Let $\\psi$ be its normal derivative at the boundary. Scaling considerations lead one to expect that the $L^2$ norm of $\\psi$ will grow as $\\lambda^{1/2}$ as $\\lambda \\to \\infty$. We prove an upper bound of the form $\\|\\psi \\|_2^2 \\leq C\\lambda$ for any Riemannian manifold, and a lower bound $c \\lambda \\leq \\|\\psi \\|_2^2$ provided that $M$ has no trapped geodesics (see the main Theorem for a precise statement). Here $c$ and $C$ are positive constants that depend on $M$, but not on $\\lambda$. The proof of the upper bound is via a Rellich-type estimate and is rather simple, while the lower bound is proved via a positive commutator estimate.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-04-08T06:58:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Pxx"
    ],
    "keywords": [
      "lower bound",
      "normal derivative",
      "upper bound",
      "compact riemannian manifold",
      "positive commutator estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......2140H"
    }
  }
}