## Abstract

Suppose that M is a compact Riemannian manifold with boundary and u is an L^{2}-normalized Dirichlet eigenfunction with eigenvalue λ. Let Ψ be its normal derivative at the boundary. Scaling considerations lead one to expect that the L^{2} norm of Ψ will grow as λ^{1/2} as λ → ∞. We prove an upper bound of the form ∥Ψ∥_{2}^{2} ≤ Cλ for any Riemannian manifold, and a lower bound cλ ≤ ∥ψ∥_{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 λ. 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.

Original language | English |
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Pages (from-to) | 289-305 |

Number of pages | 17 |

Journal | Mathematical Research Letters |

Volume | 9 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - 2002 |