Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian
Hua Chen, Hongge Chen, Hongge Chen, Hongge Chen · Discrete and Continuous Dynamical Systems · 2021
Let \begin{document}$ \Omega\subset\mathbb{R}^n \; (n\geq 2) $\end{document} be a bounded domain with continuous boundary \begin{document}$ \partial\Omega $\end{document} . In this paper, we study the Dirichlet eigenvalue problem of the fractional Laplacian which is restricted to \begin{document}$ \Omega $\end{document} with \begin{document}$ 0 . Denoting by \begin{document}$ \lambda_{k} $\end{document} the \begin{document}$ k^{th} $\end{document} Dirichlet eigenvalue of \begin{document}$ (-\triangle)^{s}|_{\Omega} $\end{document} , we establish the explicit upper bounds of the ratio \begin{document}$ \frac{\lambda_{k+1}}{\lambda_{1}} $\end{document} , which have polynomially growth in \begin{document}$ k $\end{document} with optimal increase orders. Furthermore, we give the explicit lower bounds for the Riesz mean function \begin{document}$ R_{\sigma}(z) = \sum_{k}(z-\lambda_{k})_{+}^{\sigma} $\end{document} with \begin{document}$ \sigma\geq 1 $\end{document} and the trace of the Dirichlet heat kernel of fractional Laplacian.