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A remark on Pólya’s conjecture at low frequencies

  • Pedro Freitas
Article
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Abstract

We show that the Faber–Krahn inequality implies Pólya’s conjecture for eigenvalues \(\lambda _{k}\) of the Dirichlet Laplacian in \(\mathbb {R}^n\) up to \(k = \lfloor b(n) \rfloor \), where b(n) is a function with exponential growth on the dimension. This function also appears in Pleijel’s bound for the number of nodal domains of the sequence of eigenfunctions of the ball and we improve on previous estimates by providing precise upper and lower bounds for b which coincide up to the first four terms in the expansion of \(\log (b(n))\) for large n.

Keywords

Dirichlet Laplacian Eigenvalues Faber–Krahn inequality Pólya’s conjecture 

Mathematics Subject Classification

Primary 35P15 Secondary 35J05 35J25 35P20 

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Authors and Affiliations

  1. 1.Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  2. 2.Grupo de Física Matemática, Faculdade de CiênciasUniversidade de LisboaCampo GrandePortugal

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