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Proof of the Pólya conjecture

In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain $Ω$ in $\mathbb{R}^n$. It is well known that the $k$-th Dirichlet eigenvalue $λ_k$ obeys the Weyl asymptotic formula, that is, \[ λ_k\sim\frac{4π^2}{(ω_n\mathrm{vol}Ω)^\frac{2}{n}}k^\frac{2}{n}\qquad\hbox{as}\quad k\rightarrow\infty, \] where $\mathrm{vol}Ω$ is the volume of $Ω$. In view of the above formula, Pólya conjectured that \[ λ_k\gs\frac{4π^2}{(ω_n\mathrm{vol}Ω)^\frac{2}{n}}k^\frac{2}{n}\qquad\hbox{for}\quad k\in\mathbb{N}. \] 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 by Li and Yau [Comm. Math. Phys. 88 (1983), 309--318]. They solved partially the conjecture of Pólya 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.