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

Abstract

Let $e_{ł}(x)$ be an eigenfunction with respect to the Dirichlet Laplacian ${\Delta }_N$ on a compact Riemannian manifold $N$ with boundary: ${\Delta }_N{e}_{ł}={ł}^2{e}_{ł}$ 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 $ł\ge 1$, there holds $ł\Vert e_{ł}{\Vert }_{\infty }/C\le \Vert \nabla e_{ł}{\Vert }_{\infty }\le Cł\Vert e_{ł}{\Vert }_{\infty }$, 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.

9 pages, submitted