论文标题
双曲机平面三角式图上的steklov问题
The Steklov problem on triangle-tiling graphs in the hyperbolic plane
论文作者
论文摘要
我们介绍了一个图$γ$，该图$γ$大致对双曲线平面，并研究了$γ$的边界$ω$的子图的steklov特征值。对于$（ω_l）_ {l \ geq 1} $ $γ$的子序列序列，使得$ |ω_l| \ longrightArrow \ infty $，我们证明，对于每个$ k \ in \ mathbb {n} $，$ k^{\ mbox {th}} $ eigenvalue倾向于$ 0 $ $ 0 $至$ 1/| b_l | $。证明的想法是找到双曲线平面的有界域$ n $，该域$ n $大致等距至$ω$，从而为$ n $的steklov特征值提供了上限，并通过称为离散化的过程将其传输到$ω$。
We introduce a graph $Γ$ which is roughly isometric to the hyperbolic plane and we study the Steklov eigenvalues of a subgraph with boundary $Ω$ of $Γ$. For $(Ω_l)_{l\geq 1}$ a sequence of subraphs of $Γ$ such that $|Ω_l| \longrightarrow \infty$, we prove that for each $k \in \mathbb{N}$, the $k^{\mbox{th}}$ eigenvalue tends to $0$ proportionally to $1/|B_l|$. The idea of the proof consists in finding a bounded domain $N$ of the hyperbolic plane which is roughly isometric to $Ω$, giving an upper bound for the Steklov eigenvalues of $N$ and transferring this bound to $Ω$ via a process called discretization.
