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.

Key words: The (fractional) Laplace operator, The Dirichlet eigenvalue, Higher eigenvalues, The Weyl asymptotic formula, Conjecture of Pólya, The Berezin-Li-Yau inequality, The moment of inertia