Pólya conjecture for the Neumann eigenvalues

Abstract

For a given bounded domain $\Omega \subset {\mathbb{R}}^n$ with $C^1$-smooth boundary, we prove the Pólya conjecture for the Neumann eigenvalues. In other words, we prove that \begin{eqnarray*} μ_{k+1}\le \frac{(2π)^2k^{2/n}}{(ω_n \cdot \mbox{vol}\, (Ω))^{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 $\Omega$.

7 pages. This paper has been withdrawn by the author due to an error in page 3