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

TL;DR

This paper establishes sharp gradient bounds for Dirichlet eigenfunctions on compact manifolds with boundary, linking the gradient norm to the eigenvalue and supremum norm of the eigenfunction.

Contribution

It provides explicit gradient estimates for Dirichlet eigenfunctions on manifolds with boundary, utilizing geometric properties of nodal sets and elliptic estimates, which was not previously detailed.

Findings

Gradient estimates proportional to eigenvalue

Bounds depend only on the manifold's geometry

Uses geometric properties of nodal sets and elliptic theory

Abstract

Let $e_\l(x)$ be an eigenfunction with respect to the Dirichlet Laplacian $\Delta_N$ on a compact Riemannian manifold $N$ with boundary: $\Delta_N e_\l=\l^2 e_\l$ in the interior of $N$ and $e_\l=0$ on the boundary of $N$. We show the following gradient estimate of $e_\l$: for every $\l\geq 1$, there holds $\l\|e_\l\|_\infty/C\leq \|\nabla e_\l\|_\infty\leq C{\l}\|e_\l\|_\infty$, 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.