Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866912069565022208 |
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| author | Tschanz, Léonard |
| author_facet | Tschanz, Léonard |
| contents | We study the Steklov problem on a subgraph with boundary $(Ω,B)$ of a polynomial growth Cayley graph $Γ$. 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 $Γ$. The method consists in associating a manifold $M$ to $Γ$ and a bounded domain $N \subset M$ to a subgraph $(Ω, B)$ of $Γ$. We find upper bounds for the Steklov spectrum of $N$ and transfer these bounds to $(Ω, B)$ by discretizing $N$ and using comparison Theorems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_04402 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs Tschanz, Léonard Differential Geometry We study the Steklov problem on a subgraph with boundary $(Ω,B)$ of a polynomial growth Cayley graph $Γ$. 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 $Γ$. The method consists in associating a manifold $M$ to $Γ$ and a bounded domain $N \subset M$ to a subgraph $(Ω, B)$ of $Γ$. We find upper bounds for the Steklov spectrum of $N$ and transfer these bounds to $(Ω, B)$ by discretizing $N$ and using comparison Theorems. |
| title | Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2101.04402 |