Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian

Abstract

This article is to analyze certain bounds for the sums of eigenvalues of the Dirichlet fractional Laplacian operator $(-\Delta)^{\alpha/2}|_{\Omega}$ restricted to a bounded domain $\Omega\subset{\mathbb R}^d$ with $d=2,$ $1\leq \alpha\leq 2$ and $d\geq 3,$ $0< \alpha\le 2$. A primary topic is the refinement of the Berezin-Li-Yau inequality for the fractional Laplacian eigenvalues. Our result advances the estimates recently established by Wei, Sun and Zheng in [34]. Another aspect of interest in this work is that we obtain some estimates for the sums of powers of the eigenvalues of the fractional Laplacian operator.