On the Pólya conjecture for circular sectors and for balls

In 1954, G. Polya conjectured that the counting function $N(\mathrm{\Omega },\mathrm{\Lambda })$ of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set $\mathrm{\Omega }\subset R^d$ is lesser (resp. greater) than $(2\pi {)}^{-d}{\omega }_d\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }{\mathrm{\Lambda }}^{d\mathrm{/}2}$. Here $\mathrm{\Lambda }$ is the spectral parameter, and ${\omega }_d$ is the volume of the unit ball. We prove this conjecture for both Dirichlet and Neumann boundary problems for any circular sector, and for the Dirichlet problem for a ball of arbitrary dimension. We heavily use the ideas from \cite{LPS}.