Proof of the Pólya conjecture

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 $\mathbb{R}^n$. It is well known that the $k$-th Dirichlet eigenvalue $\lambda_k$ obeys the Weyl asymptotic formula, that is, [ \lambda_k\sim\frac{4\pi^2}{(\omega_n\mathrm{vol}\Omega)^\frac{2}{n}}k^\frac{2}{n}\qquad\hbox{as}\quad k\rightarrow\infty, ] where $\mathrm{vol}\Omega$ is the volume of $\Omega$. In view of the above formula, P\'{o}lya conjectured that [ \lambda_k\gs\frac{4\pi^2}{(\omega_n\mathrm{vol}\Omega)^\frac{2}{n}}k^\frac{2}{n}\qquad\hbox{for}\quad k\in\mathbb{N}. ] This is the well-known conjecture of P\'{o}lya. 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 by Li and Yau [Comm. Math. Phys. 88 (1983), 309--318]. They solved partially the conjecture of P\'{o}lya with a slight difference by a factor $n/(n+2)$. Here, following the argument of Li and Yau on the whole, we shall thoroughly solve the above conjecture.