Gradient estimate of a Dirichlet eigenfunction on a compact manifold with boundary

Let $e_ł(x)$ be an eigenfunction with respect to the Dirichlet Laplacian $Δ_N$ on a compact Riemannian manifold $N$ with boundary: $Δ_N e_ł=ł^2 e_ł$ in the interior of $N$ and $e_ł=0$ on the boundary of $N$. We show the following gradient estimate of $e_ł$: for every $ł\geq 1$, there holds $ł\|e_ł\|_\infty/C\leq \|\nabla e_ł\|_\infty\leq Cł\|e_ł\|_\infty$, 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.

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