  Let $e_\l(x)$ be an eigenfunction with respect to the Dirichlet Laplacian
$\Delta_N$ on a compact Riemannian manifold $N$ with boundary: $\Delta_N
e_\l=\l^2 e_\l$ in the interior of $N$ and $e_\l=0$ on the boundary of $N$. We
show the following gradient estimate of $e_\l$: for every $\l\geq 1$, there
holds $\l\|e_\l\|_\infty/C\leq \|\nabla e_\l\|_\infty\leq
C{\l}\|e_\l\|_\infty$, where $C$ is a positive constant depending only on $N$.
In the proof, we use a basic geometrical property of nodal sets of
eigenfunctions and elliptic apriori estimates.
