Gradient Estimates on Dirichlet Eigenfunctions

By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants $c\_1(D)$ and $c\_2(D)$ 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 $\phi$ of $-\Delta$ with eigenvalue $\lambda$. In particular, when $D$ is convex with nonnegative Ricci curvature, this estimate holds for $c\_1(D)=\frac{1}{de}$ and $c\_2(D)=\sqrt{e}\left(\frac{\sqrt{2}}{\sqrt{\pi}}+\frac{\sqrt{\pi}}{4\sqrt{2}}\right)$. Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper.