We study the Steklov problem on a subgraph with boundary \((\Omega ,B)\) of a polynomial growth Cayley graph \(\Gamma\). For \((\Omega _l, B_l)_{l=1}^\infty\) a sequence of subgraphs of \(\Gamma\) such that \(|\Omega _l| \longrightarrow \infty\), we prove that for each \(k \in {\mathbb {N}}\), the kth eigenvalue tends to 0 proportionally to \(1/|B|^{\frac{1}{d-1}}\), where d represents the growth rate of \(\Gamma\). The method consists in associating a manifold M to \(\Gamma\) and a bounded domain \(N \subset M\) to a subgraph \((\Omega , B)\) of \(\Gamma\). We find upper bounds for the Steklov spectrum of N and transfer these bounds to \((\Omega , B)\) by discretizing N and using comparison theorems.

我们研究了多项式增长凯莱图\(\Gamma\) 的边界为\((\Omega ,B)\)的子图上的 Steklov 问题。对于\((\Omega _l, B_l)_{l=1}^\infty\) \(\Gamma\)的一系列子图使得\(|\Omega _l| \longrightarrow \infty\)，我们证明对于每个\(k \in {\mathbb {N}}\)，第k个特征值趋于 0 与\(1/|B|^{\frac{1}{d-1}}\)成比例，其中d表示\(\Gamma\)的增长率。该方法包括将流形M与\(\Gamma\)和有界域\(N \subset M\) 相关联到子图\（（\欧米茄，B）\）的\（\伽玛\） 。我们找到N的 Steklov 谱的上限，并通过离散N和使用比较定理将这些边界转移到\((\Omega , B)\)。