The weak Pleijel theorem with geometric control

Pierre Bérard (1) , Bernard Helffer (2, 3)

1 IF - Institut Fourier
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 LMJL - Laboratoire de Mathématiques Jean Leray

Let $\Omega\subset \mathbb R^d\,, d\geq 2$, be a bounded open set, and denote by $\lambda_j(\Omega), j\geq 1$, 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 $\lambda_j(\Omega)$, for which there exists an associated eigenfunction with precisely $j$ 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 $\Omega$. We will see that this is connected with one of the favorite problems considered by Y. Safarov.

Mots clés: Dirichlet Laplacian; Nodal domains; Courant theorem; Pleijel theorem

Domaines: Théorie spectrale [math.SP]; Physique mathématique [math-ph]; Géométrie différentielle [math.DG]

Pierre Bérard, Bernard Helffer. The weak Pleijel theorem with geometric control. Journal of Spectral Theory, 2016, 6 (4), pp.717--733. ⟨hal-01244630v2⟩