# Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders

Abstract : The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, Möbius strips,\ldots . A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic $0$, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders $(0,\pi) \times \mathbb{S}^1_r$ where $r \in \{0.5,1\}$ is the radius of the circle $\mathbb{S}^1_r$, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues.
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Cited literature [28 references]

https://hal.archives-ouvertes.fr/hal-02937338
Contributor : Pierre Bérard <>
Submitted on : Wednesday, September 16, 2020 - 3:58:00 PM
Last modification on : Sunday, November 15, 2020 - 3:25:38 AM

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### Identifiers

• HAL Id : hal-02937338, version 2
• ARXIV : 2007.09219

### Citation

Pierre Bérard, Bernard Helffer, Rola Kiwan. Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders. 2020. ⟨hal-02937338v2⟩

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