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

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)}).