Eigenvalues of the Laplacian on Riemannian manifolds

TL;DR

This paper investigates eigenvalues of the Laplacian on bounded domains within Riemannian manifolds, deriving inequalities that extend previous results for lower order eigenvalues using orthonormal basis properties.

Contribution

It introduces new eigenvalue inequalities for the Laplacian on Riemannian manifolds, expanding upon prior work by utilizing the orthonormal basis of eigenfunctions.

Findings

Derived inequalities for Laplacian eigenvalues on Riemannian manifolds.

Extended results of Chen and Cheng for lower order eigenvalues.

Utilized orthonormal basis of eigenfunctions instead of Rayleigh-Ritz formula.

Abstract

For a bounded domain $\Omega$ with a piecewise smooth boundary in a complete Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of $L^2(\Omega)$ in place of the Rayleigh-Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng \cite{CC}.