In 1954, G. Polya conjectured that the counting function $N(\Omega,\Lambda)$ of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set $\Omega\subset R^d$ is lesser (resp. greater) than $(2\pi)^{ -d} \omega_d |\Omega| \Lambda^{ d/2}$. Here $\Lambda$ 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}.
Categories: math-ph math.MP math.SP