Abstract
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 will grow as $\lamdba^(1/2)$ as $\lamda \rightarrow \infty$. We sketch proofs of an upper bound of the form $\Vert \psi |Vert_2^2 \leq C\lambda$ for any Riemannian manifold, and a lower bound $c\lamda \leq \Vert \psi \Vert_2^2$ 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 $\lamda$. Full details will appear in [3].
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