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VOL. 41 | 2003 $L^2$ bounds for normal derivatives of Dirichlet eigenfunctions
Andrew Hassell, Terence Tao

Editor(s) Xuan Thinh Duong, Alan Pryde

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 will grow as $\lamdba^(1/2)$ as $\lamda \rightarrow \infty$. We sketch proofs of an upper bound of the form $\Vert \psi |Vert_2^2 \leq C\lambda$ for any Riemannian manifold, and a lower bound $c\lamda \leq \Vert \psi \Vert_2^2$ 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 $\lamda$. Full details will appear in [3].

Information

Published: 1 January 2003
First available in Project Euclid: 18 November 2014

zbMATH: 1106.58023
MathSciNet: MR1994515

Rights: Copyright © 2003, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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