  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.
