Gradient Estimates on Dirichlet Eigenfunctions

By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants $c\mathrm{\_}1(D)$ and $c\mathrm{\_}2(D)$ for a $d$-dimensional compact Riemannian manifold $D$ with boundary such that $c_1(D)\sqrt{\lambda}|\phi|_\infty \le |\nabla \phi|_\infty\le c_2(D)\sqrt{\lambda} |\phi|_\infty$ holds for any Dirichlet eigenfunction $\varphi$ of $-\mathrm{\Delta }$ with eigenvalue $\lambda$. In particular, when $D$ is convex with nonnegative Ricci curvature, this estimate holds for $c\mathrm{\_}1(D)=\frac{1}{de}$ and $c\mathrm{\_}2(D)=\sqrt{e}(\frac{\sqrt{2}}{\sqrt{\pi }}+\frac{\sqrt{\pi }}{4\sqrt{2}})$. Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper.