Lower Bounds for Laplacian and Fractional Laplacian Eigenvalues

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].