论文标题
等级上限是离散steklov问题的第一个特征值
Isoperimetric upper bound for the first eigenvalue of discrete Steklov problems
论文作者
论文摘要
我们研究了具有边界有限图上定义的steklov问题的第一个非零特征值的上限。对于与一组多项式增长相关的Cayley图中包含边界的有限图，我们根据图和边界的顶点的数量为第一个非零steklov特征值提供了上限。作为推论，如果具有边界的图还满足离散的等值不平等，我们表明，随着图形的顶点趋向于无穷大，第一个非零steklov特征值趋于零。这扩展了Han和Hua的最新结果，他们在$ \ Mathbb {z}^n $的情况下获得了类似的结果。我们使用与多项式生长组相关的Cayley图的度量特性获得了结果。
We study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices of the graph and of its boundary. As a corollary, if the graph with boundary also satisfies a discrete isoperimetric inequality, we show that the first non-zero Steklov eigenvalue tends to zero as the number of vertices of the graph tends to infinity. This extends recent results of Han and Hua, who obtained a similar result in the case of $\mathbb{Z}^n$. We obtain the result using metric properties of Cayley graphs associated to groups of polynomial growth.
