标题： Dirichlet-to-Neumann算子特征函数的Courant型节点域界限不存在 显示英文标题

标题： Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator

摘要： 给定一个维数为$n\geq 3$的具有边界的紧致流形$\mathcal M$，以及任何整数$K$和$N$，我们证明在$\mathcal M$上存在一种度量，使得在$\partial\mathcal M$上的狄利克雷-诺伊曼映射的前$K$个非常数特征函数至少有$N$个节点成分。 这给出了对一个疑问的否定回答，即Dirichlet-to-Neumann特征函数的节点域数量是否满足Courant型界，该问题最近在Girouard和Polterovich [21, 开放问题9]以及Colbois、Girouard、Gordon和Sher [9, 开放问题10.14]的综述中被提及。 显示英文摘要

摘要： Given a compact manifold $\mathcal M$ with boundary of dimension $n\geq 3$ and any integers $K$ and $N$, we show that there exists a metric on $\mathcal M$ for which the first $K$ nonconstant eigenfunctions of the Dirichlet-to-Neumann map on $\partial\mathcal M$ have at least $N$ nodal components. This provides a negative answer to the question of whether the number of nodal domains of Dirichlet-to-Neumann eigenfunctions satisfies a Courant-type bound, which has been featured in recent surveys by Girouard and Polterovich [21, Open problem 9] and by Colbois, Girouard, Gordon and Sher [9, Open question 10.14].