# Courant-sharp Robin eigenvalues for the square -- the case with small Robin parameter--

## TL;DR

This paper investigates Courant-sharp Robin eigenvalues for a square domain, focusing on the case where the Robin parameter is small, extending previous results from Dirichlet and Neumann cases to small Robin parameters.

## Contribution

It extends the analysis of Courant-sharp Robin eigenvalues for the square to the case of small Robin parameters, building on prior work for large parameters and Neumann conditions.

## Key findings

- Characterization of Courant-sharp eigenvalues for small Robin parameter h
- Extension of previous results from Dirichlet and Neumann cases
- Insights into the nodal structure behavior as h approaches zero

## Abstract

This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter $h$). For the square, our first paper focused on the case where $h$ is large and extended results that were obtained by Pleijel, B\'erard--Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of $h$, where $h$ is positive and not close to $0$. In this second paper, we extend results that were obtained by Helffer--Persson-Sundqvist for the Neumann problem to the case where $h>0$ is small.