Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions

Suppose that M is a compact Riemannian manifold with boundary and u is an L²-normalized Dirichlet eigenfunction with eigenvalue λ. Let Ψ be its normal derivative at the boundary. Scaling considerations lead one to expect that the L² norm of Ψ will grow as λ^1/2 as λ → ∞. We prove an upper bound of the form ∥Ψ∥₂² ≤ Cλ for any Riemannian manifold, and a lower bound cλ ≤ ∥ψ∥₂² 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.