Universal Bounds for Fractional Laplacian on a Bounded Open Domain in $\mathbb{R}^{n}$

TL;DR

This paper establishes universal bounds for eigenvalues of the fractional Laplacian on bounded domains, extending previous inequalities for polyharmonic operators and considering eigenvalues with lower order in the same setting.

Contribution

It introduces a new universal eigenvalue inequality for the fractional Laplacian that generalizes existing bounds for polyharmonic operators, including lower order eigenvalues.

Findings

Derived a universal eigenvalue inequality for fractional Laplacian.

Extended eigenvalue bounds to lower order eigenvalues.

Generalized bounds for polyharmonic operators to fractional case.

Abstract

Let $\Omega$ be a bounded open domain on the Euclidean space $\mathbb{R}^{n}$ and $\mathbb{Q}_{+}$ be the set of all positive rational numbers. In 2017, Chen and Zeng investigated the eigenvalues with higher order of the fractional Laplacian $\left.(-\Delta)^{s}\right|_{\Omega}$ for $s>0$ and $s \in \mathbb{Q}_{+}$, and they obtained a universal inequality of Yang type(\emph{ Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain, Calculus of Variations and Partial Differential Equations, (2017) \textbf{56}:131}). In the spirit of Chen and Zeng's work, we study the eigenvalues of fractional Laplacian, and establish an inequality of eigenvalues with lower order under the same condition. Also, our eigenvalue inequality is universal and generalizes the eigenvalue inequality for the poly-harmonic operators given by Jost et al.(\emph{Universal bounds for eigenvalues of polyharmonic operator. Trans. Amer. Math. Soc. {\bf 363}(4), 1821-1854 (2011)}).