An improved version of a spectral inequality by Payne

TL;DR

This paper improves Payne's classical spectral inequality by providing a sharper bound on the first eigenvalue of the Dirichlet Laplacian, advancing the understanding of spectral inequalities.

Contribution

The paper offers a refined, quantitative version of Payne's inequality, enhancing the classical spectral bound with a sharper estimate.

Findings

Improved spectral bound for the Dirichlet Laplacian eigenvalue

Establishment of a more precise inequality than Payne's original

Potential for further quantitative enhancements in spectral theory

Abstract

A celebrated inequality by Payne relates the first eigenvalue of the Dirichlet Laplacian to the first eigenvalue of the buckling problem. Motivated by the goal of establishing a quantitative version of this inequality, we show that Payne's original estimate - which is not sharp - can in fact be improved. Our result provides a refined spectral bound and opens the way to further investigations into quantitative enhancements of classical inequalities in spectral theory.