Lower Bounds for Laplacian and Fractional Laplacian Eigenvalues

TL;DR

This paper derives sharper lower bounds for the sums of eigenvalues of both the Laplacian and fractional Laplacian operators on bounded domains, improving upon previous results in spectral theory.

Contribution

It introduces improved lower bounds for eigenvalue sums of Laplacian and fractional Laplacian, advancing spectral bounds in mathematical analysis.

Findings

Sharper lower bounds for Laplacian eigenvalues

Enhanced bounds for fractional Laplacian eigenvalues

Improved spectral inequalities compared to prior work

Abstract

In this paper, we investigate eigenvalues of Laplacian on a bounded domain in an $n$-dimensional Euclidean space and obtain a sharper lower bound for the sum of its eigenvalues, which gives an improvement of results due to A. D. Melas [15]. On the other hand, for the case of fractional Laplacian $(-\Delta)^{\alpha/2}|_{D}$, where $\alpha\in(0,2]$, we obtain a sharper lower bound for the sum of its eigenvalues, which gives an improvement of results due to S.Y. Yolcu and T. Yolcu [23].