Proof of the P\'{o}lya conjecture

In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain $\mathrm{\Omega }$ in ${\mathbb{R}}^n$. It is well known that the $k$-th Dirichlet eigenvalue ${\lambda }_k$ obeys the Weyl asymptotic formula, that is, $${\lambda }_k\sim \frac{4{\pi }^2}{({\omega }_n\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{\Omega }{)}^{\frac{2}{n}}}{k}^{\frac{2}{n}}\text{as}k\to \mathrm{\infty },$$ where $\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{\Omega }$ is the volume of $\mathrm{\Omega }$. In view of the above formula, P'{o}lya conjectured that $${\lambda }_k\text{gs}\frac{4{\pi }^2}{({\omega }_n\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{\Omega }{)}^{\frac{2}{n}}}{k}^{\frac{2}{n}}\text{for}k\in \mathbb{N}\mathrm{.}$$ 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 $n\mathrm{/}(n+2)$. Here, following the argument of Li and Yau on the whole, we shall thoroughly solve the above conjecture.