Erratum for "Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions"

By Andrew Hassell, Terence Tao. 2002

77 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1694933132130ae0dbf66313bdc8d7a8fc0cbae7

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In Computational Theory and Mathematics

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Citations in the corpus

Inverse problems for the connection Laplacian (2015, unknown): “In [28] a similar difficulty is treated by imposing an asymptotic spectral condition of the type that was first studied in [19].”
Geometrical Structure of Laplacian Eigenfunctions (2012, background): “where the upper bound holds for any Riemannian manifold, while the lower bound is valid provided that M has no trapped geodesics [177].”
Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets (2012, methodology): “Exact controllability and the Hassell-Tao condition.”