# Courant-sharp Robin eigenvalues for the square and other planar domains

## TL;DR

This paper investigates when Courant's nodal domain theorem achieves equality for Robin eigenvalues on squares and planar domains, extending previous results and analyzing the large Robin parameter case.

## Contribution

It extends known results on Courant-sharp eigenvalues to Robin boundary conditions, especially for large Robin parameters, and provides semi-stability results for nodal domains.

## Key findings

- Partial extension of Courant-sharp eigenvalue results to Robin conditions on squares.
- Identification of eigenvalue cases where equality holds for large Robin parameters.
- Semi-stability results for the number of nodal domains as Robin parameter varies.

## Abstract

This paper is devoted to the determination of the cases where there is equality in Courant's nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, B\'erard--Helffer, Helffer--Persson--Sundqvist for the Dirichlet and Neumann problems. After proving some general results that hold for any value of the Robin parameter $h$, we focus on the case when $h$ is large. We hope to come back to the analysis when $h$ is small in a second paper. We also obtain some semi-stability results for the number of nodal domains of a Robin eigenfunction of a domain with $C^{2,\alpha}$ boundary ($\alpha >0$) as $h$ large varies.