### Refined Lower Bound for Sums of Eigenvalues of the Laplace Operator

Yue He1,3,*(),Qihua Ruan2,3()

1. 1Institute of Mathematics, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210023
2Department of Mathematics, Putian University, Fujian Putian 351100
3Key Laboratory of Applied Mathematics (Putian University), Fujian Province University, Fujian Putian 351100
• Received:2022-03-13 Revised:2022-08-15 Online:2023-02-26 Published:2023-03-07
• Supported by:
The NSFC(11871278);The NSFC(11971253);Key Laboratory of Applied Mathematics of Fujian Province University(Putian University)(SX202101)

Abstract:

In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain $\Omega$ in $\Bbb R ^n$. It is well known that the $k$-th Dirichlet eigenvalue $\lambda_k(\Omega)$ obeys the Weyl asymptotic formula, that is, $\lambda_k(\Omega)\sim\frac{4\pi^2}{[\omega_nV(\Omega)]^\frac{2}{n}}k^\frac{2}{n} \qquad\hbox{as}\quad k\rightarrow\infty,$ where $\omega_n$ and $V(\Omega)$ are the volume of $n$-dimensional unit ball in $\Bbb R ^n$ and the volume of $\Omega$ respectively. In view of the above formula, Pólya conjectured that $\lambda_k(\Omega)\geq\frac{4\pi^2}{[\omega_nV(\Omega)]^\frac{2}{n}}k^\frac{2}{n} \qquad\hbox{for}\quad k\in{\Bbb N}.$ This is the well-known conjecture of Pólya. Studies on this topic have a long history with much work. In particular, one of the more remarkable achievements in recent tens years has been achieved independently by Berezin[2] and Li and Yau[4], respectively. They solved partially the conjecture of Pólya with a slight difference by a factor $n/(n+2)$. Later, Melas[7] improved Berezin-Li-Yau's estimate by adding an additional positive term of the order of $k$ to the right side. Here, following almost the same argument as Melas, we further refine Melas's estimate.

CLC Number:

• O186.1
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