{
  "id": "1004.2517",
  "version": "v4",
  "published": "2010-04-14T22:18:08.000Z",
  "updated": "2011-06-16T20:57:19.000Z",
  "title": "Upper and lower bounds for normal derivatives of spectral clusters of Dirichlet Laplacian",
  "authors": [
    "Xiangjin Xu"
  ],
  "comment": "14 pages, Rewrite section 6 (the appendix) and correct a mistake in section 6",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "In this paper, we prove the upper and lower bounds for normal derivatives of spectral clusters $u=\\chi_{\\lambda}^s f$ of Dirichlet Laplacian $\\Delta_M$, $$c_s \\lambda\\|u\\|_{L^2(M)} \\leq \\| \\partial_{\\nu}u \\|_{L^2(\\partial M)} \\leq C_s \\lambda \\|u\\|_{L^2(M)} $$ where the upper bound is true for any Riemannian manifold, and the lower bound is true for some small $0<s<s_M$, where $s_M$ depends on the manifold only, provided that $M$ has no trapped geodesics (see Theorem \\ref{Thm3} for a precise statement), which generalizes the early results for single eigenfunctions by Hassell and Tao.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-06-16T20:57:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P20",
      "35J25",
      "58J40",
      "35R01"
    ],
    "keywords": [
      "lower bound",
      "normal derivatives",
      "spectral clusters",
      "dirichlet laplacian",
      "precise statement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.2517X"
    }
  }
}