P\'olya conjecture for the Neumann eigenvalues
  For a given bounded domain $\Omega\subset {\Bbb R}^n$ with $C^1$-smooth boundary, we prove the P\'olya conjecture for the Neumann eigenvalues. In other words, we prove that \begin{eqnarray*} \mu_{k+1}\le \frac{(2\pi)^2k^{2/n}}{(\omega_n \cdot \mbox{vol}\, (\Omega))^{2/n}} \quad \;\; \mbox{for all} \;\; k=0,1,2,3,\cdots,\end{eqnarray*} where $\mu_k$ is the $k$-th Neumann eigenvalue of the Laplacian for $\Omega$. 
Genqian Liu
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