On the number of Courant-sharp Dirichlet eigenvalues

Abstract

We consider arbitrary open sets $\Omega$ in Euclidean space with finite Lebesgue measure, and obtain upper bounds for (i) the largest Courant-sharp Dirichlet eigenvalue of $\Omega$, (ii) the number of Courant-sharp Dirichlet eigenvalues of $\Omega$. This extends recent results of P. Bérard and B. Helffer.

arXiv comment10 pages, 2 figures