Gradient Estimates on Dirichlet Eigenfunctions

TL;DR

This paper derives explicit gradient bounds for Dirichlet and Neumann eigenfunctions on compact Riemannian manifolds using stochastic analysis, providing concrete constants and estimates especially for convex manifolds with nonnegative Ricci curvature.

Contribution

It introduces explicit constants for gradient estimates of Dirichlet eigenfunctions and extends similar bounds to Neumann eigenfunctions on Riemannian manifolds.

Findings

Explicit constants for gradient bounds on Dirichlet eigenfunctions.

Gradient estimates for Neumann eigenfunctions.

Special case bounds for convex manifolds with nonnegative Ricci curvature.

Abstract

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.