The weak Pleijel theorem with geometric control

Abstract

Let $\Omega \subset {\mathbb{R}}^d,d\ge 2$, be a bounded open set, and denote by $\lambda \_j(\Omega ),j\ge 1$, 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 $\lambda \_j(\Omega )$, 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.

Revised Oct. 12, 2016. To appear in Journal of Spectral Theory 6 (2016)

References in corpus (1)

• Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle

Cited by in corpus (1)

• Courant-sharp eigenvalues of Neumann 2-rep-tiles