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

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Let eł(x) be an eigenfunction with respect to the Dirichlet Laplacian Δₙ on a compact Riemannian manifold N with boundary: Δₙ eł=ł² eł in the interior of N and eł=0 on the boundary of N. We show the following gradient estimate of eł: for every ł≥ 1, there holds łeł∞/C≤ ∇ eł∞≤ Cłeł∞, 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.