Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions

Details Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions

2002-02-15

arxiv.org/abs/math/0202140v2

Mathematics

Analysis of PDEs

16 pages, 1 figure. Some minor errors and ambiguous notation corrected, and the diagram compressed

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-709701