# On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains--with some numerics--

## TL;DR

This paper investigates the multiplicity of the second eigenvalue of the Laplacian in non-simply connected domains, providing numerical analysis and revisiting a specific example with multiplicity three.

## Contribution

It offers a detailed numerical and theoretical analysis of eigenvalue multiplicities in complex domains, including revisiting a notable example and examining eigenvalues with symmetric cracks.

## Key findings

- Second eigenvalue multiplicity can be three in certain non-simply connected domains.
- Eigenvalues of the Laplacian are sensitive to domain geometry and boundary modifications.
- Numerical methods reveal eigenvalue behavior in domains with symmetric cracks.

## Abstract

We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity three. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a smaller concentric disk in function of their size.