Lower bounds for fractional Laplacian eigenvalues

Abstract

In this paper, we investigate eigenvalues of fractional Laplacian (–Δ)α/2|D, where α ∈ (0, 2], on a bounded domain in an n-dimensional Euclidean space and obtain a sharper lower bound for the sum of its eigenvalues, which improves some results due to Yildirim Yolcu and Yolcu in [Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Commun. Contemp. Math. 15(3) (2013) 1250048]. In particular, for the case of Laplacian, we obtain a sharper eigenvalue inequality, which gives an improvement of the result due to Melas in [A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003) 631–636].