Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions
| dc.creator | Hassell, Andrew | |
| dc.creator | Tao, Terence | |
| dc.date | 2002-02-15 | |
| dc.date | 2002-04-08 | |
| dc.date.accessioned | 2026-07-07T04:46:28Z | |
| dc.date.available | 2026-07-07T04:46:28Z | |
| dc.description | 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 $λ\to \infty$. We prove an upper bound of the form $\|ψ\|_2^2 \leq Cλ$ for any Riemannian manifold, and a lower bound $c λ\leq \|ψ\|_2^2$ 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. | |
| dc.description | 16 pages, 1 figure. Some minor errors and ambiguous notation corrected, and the diagram compressed | |
| dc.identifier | https://arxiv.org/abs/math/0202140 | |
| dc.identifier | http://arxiv.org/abs/math/0202140 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63342 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Spectral Theory | |
| dc.subject | 35Pxx | |
| dc.title | Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions | |
| dc.type | text |