JournalsjstVol. 6 , No. 4pp. 717–733

The weak Pleijel theorem with geometric control

  • Pierre Bérard

    Université Grenoble I, Saint-Martin-d'Hères, France
  • Bernard Helffer

    Université de Nantes, France
The weak Pleijel theorem with geometric control cover
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Abstract

Let ΩRd,d2\Omega \subset \mathbb R^d, d \geq 2, be a bounded open set, and denote by λj(Ω),j1\lambda_j(\Omega), j\ge 1, the eigenvalues of the Dirichlet Laplacian arranged in nondecreasing order, with multiplicities. The weak form of Pleijel's theorem states that the number of eigenvalues λj(Ω)\lambda_j(\Omega), for which there exists an associated eigenfunction with precisely jj nodal domains (Courant-sharp eigenvalues), is finite. The purpose of this note is to determine an upper bound for Courant-sharp eigenvalues, expressed in terms of simple geometric invariants of Ω\Omega. We will see that this is connected with one of the favorite problems considered by Y. Safarov.

Cite this article

Pierre Bérard, Bernard Helffer, The weak Pleijel theorem with geometric control. J. Spectr. Theory 6 (2016), no. 4 pp. 717–733

DOI 10.4171/JST/138