Upper and Lower Bounds for Normal Derivatives of Dirichlet Eigenfunctions

Suppose that M is a compact Riemannian manifold with boundary and u is an L-normalized Dirichlet eigenfunction with eigenvalue λ. Let ψ be its normal derivative at the boundary. Scaling considerations lead one to expect that the L norm of ψ will grow as λ1/2 as λ → ∞. We prove an upper bound of the form ‖ψ‖ 2 ≤ Cλ for any Riemannian manifold, and a lower bound cλ ≤ ‖ψ‖ 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 onM , 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.