Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions

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.

Locations

• Mathematical Research Letters - View - PDF
• arXiv (Cornell University) - PDF
• arXiv (Cornell University) - View
• Mathematical Research Letters - View - PDF
• arXiv (Cornell University) - PDF
• arXiv (Cornell University) - View