Let \((M,g)\) be a compact Riemannian manifold with smooth boundary \(\partial M\). Let \(u_j\) be an eigenfunction and \(\lambda_j\) the corresponding eigenvalue for the Dirichlet Laplacian. Let \(\psi_j\) be the normal derivative of \(u_j\) on \(\partial M\).
The author establishes an upper bound \[ |\psi_j|_{L^2(\partial M)}^2\leq C\lambda_j \] where \(C=C(M)\). He establishes a corresponding lower bound \[ c\lambda_j\leq |\psi_j|_{L^2(\partial M)}^2 \] where \(c=c(M)\) under the additional assumption that \(M\) can be embedded in the interior of a compact manifold \(N\) with boundary such that every geodesic in \(M\) eventually meets the boundary of \(N\); i.e. there are no ‘trapped’ geodesics. In particular, the lower bound holds if \(M\) is a sub-domain of Euclidean space. The author shows that some condition is necessary to ensure the existence of a lower bound by noting that the lower bound fails for the hemisphere of the sphere and the cylinder in \(\mathbb{R}^3\).