论文标题
紧凑型平面表面的courant-sharp特征值：klein瓶和气缸
Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders
论文作者
论文摘要
确定存在哪种特征值的特征功能的问题，该特征功能具有与相关特征值（Courant-Sharp属性）标签相同的汇总域（Courant-Sharp属性）的动机，这是通过最小光谱分区的分析来激励的。在以前的作品中，已经分析了许多示例，对应于正方形，矩形，磁盘，三角形，托里，莫比乌斯条，\ ldots。用于进一步研究的天然玩具模型是平坦的klein瓶，一种不可定向的表面，具有Euler特性$ 0 $，尤其是与Square Torus相关的Klein瓶，其特征值具有较高的倍数。在本说明中，我们证明，与正方形的圆环相关的扁平klein瓶的库兰特特征值（带正方形的基本域）是第一和第二特征值。 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.
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.
