• Corpus ID: 251403062

On the P\'olya conjecture for circular sectors and for balls

  title={On the P\'olya conjecture for circular sectors and for balls},
  author={Nikolai Filonov},
In 1954, G. P´olya conjectured that the counting function N (Ω , Λ) of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set Ω ⊂ R d is lesser (resp. greater) than (2 π ) − d ω d | Ω | Λ d/ 2 . Here Λ is the spectral parameter, and ω 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… 
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