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

TL;DR

This paper establishes a gradient estimate for Neumann eigenfunctions on compact manifolds with boundary, providing bounds on the gradient in terms of the eigenvalue and the eigenfunction's supremum norm.

Contribution

It introduces a new gradient estimate for Neumann eigenfunctions and spectral projections, extending understanding of eigenfunction behavior on manifolds with boundary.

Findings

Gradient estimate for spectral projection operators involving eigenfunctions.

Bound on the gradient of Neumann eigenfunctions proportional to eigenvalue.

Provides tools for analyzing eigenfunction regularity on manifolds with boundary.

Abstract

Let $e_\l(x)$ be a Neumann eigenfunction with respect to the positive Laplacian $\Delta$ on a compact Riemannian manifold $M$ with boundary such that $\Delta\, e_\l=\l^2 e_\l$ in the interior of $M$ and the normal derivative of $e_\l$ vanishes on the boundary of $M$. Let $\chi_\lambda$ be the unit band spectral projection operator associated with the Neumann Laplacian and $f$ a square integrable function on $M$. We show the following gradient estimate for $\chi_\lambda\,f$ as $\lambda\geq 1$: $\|\nabla\ \chi_\l\ f\|_\infty\leq C\l \|\chi_\l\f\|_\infty+\l^{-1}\|\Delta\ \chi_\l\ f\|_\infty$, where $C$ is a positive constant depending only on $M$. As a corollary, we obtain the gradient estimate of $e_\l$: for every $\l\geq 1$, there holds $\|\nabla e_\l\|_\infty\leq C\,\l\, \|e_\l\|_\infty$.