紧致平面的Courant-sharp特征值：克莱因瓶和圆柱
Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders
摘要：
确定哪些特征值存在与相关特征值的标签具有相同数目的节点域的特征函数（Courant-sharp性质）的问题是由最小谱划分的分析激发的。在以前的工作中，已经分析了与正方形、矩形、圆盘、三角形、圆环、默比乌斯带、\ldots对应的许多示例。用于进一步研究的自然玩具模型是平坦的克莱因瓶，具有欧拉示性数$0$的不可可定向曲面，并且特别是与方形环面相关联的克莱因瓶，其特征值具有较高的多重性。本文证明了与正方形环面（分别为）有关的平坦Klein瓶的Courant-sharp特征值是唯一的。具有平方基本区）是第一和第二特征值。我们还考虑了扁平圆柱$(0,π) \times \mathbb{S}^1_r$，其中$r \in \{0.5,1\}$是半径的圆$\mathbb{S}^1_r$，我们表明，只有Courant-sharp Dirichlet特征值的这些圆柱的第一和第二特征值。
英文摘要：
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 $(0,π) \times \mathbb{S}^1_r$ 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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