Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs
| Autor: | Tschanz, Léonard |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Ann Glob Anal Geom 61, 37 (2022) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10455-021-09799-w |
| Popis: | We study the Steklov problem on a subgraph with boundary $(\Omega,B)$ of a polynomial growth Cayley graph $\Gamma$. We prove that for each $k \in \mathbb{N}$, the $k^{\mbox{th}}$ 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. Comment: 17 pages |
| Databáze: | arXiv |
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