On the number of Courant-sharp Dirichlet eigenvalues

TL;DR

This paper provides upper bounds on the largest Courant-sharp Dirichlet eigenvalue and the total number of such eigenvalues for arbitrary open sets in Euclidean space, extending previous results.

Contribution

It introduces new bounds for Courant-sharp eigenvalues applicable to any open set with finite measure, broadening the scope of earlier work.

Findings

Upper bounds for the largest Courant-sharp eigenvalue

Upper bounds for the number of Courant-sharp eigenvalues

Extension of previous results to arbitrary open sets

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. B\'erard and B. Helffer.