Gradient and Hessian Estimates for Dirichlet and Neumann Eigenfunctions

TL;DR

This paper derives sharp gradient and Hessian estimates for Dirichlet and Neumann eigenfunctions on Riemannian manifolds with boundary, using integral formulas and curvature bounds.

Contribution

It introduces new integral formulas and bounds that improve understanding of eigenfunction behavior on manifolds with boundary.

Findings

Sharp gradient estimates for eigenfunctions

Precise Hessian bounds for eigenfunctions

Curvature bounds influence eigenfunction estimates

Abstract

We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the Dirichlet and Neumann eigenfunctions.