Refined Eigenvalue Bounds on the Dirichlet Fractional Laplacian

TL;DR

This paper derives sharper lower bounds for eigenvalues of the Dirichlet fractional Laplacian, improving existing inequalities and extending results to various dimensions and fractional orders.

Contribution

It introduces refined eigenvalue bounds for the Dirichlet fractional Laplacian, enhancing previous inequalities and covering a broader range of dimensions and fractional powers.

Findings

Sharper lower bounds for eigenvalue sums established

Improved Berezin-Li-Yau type inequalities derived

Extensions to various dimensions and fractional orders

Abstract

The purpose of this article is to establish new lower bounds for the sums of powers 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 $d\geq 3,$ $0< \alpha\le 2$. Our main result yields a sharper lower bound, in the sense of Weyl asymptotics, for the Berezin-Li-Yau type inequality improving the previous result in [36]. Furthermore, we give a result improving the bounds for analogous elliptic operators in [19].