arXiv subjects

Léonard Tschanz

Publications and source records attributed to Léonard Tschanz.

4 recordsLinked to original sources

arXiv2024

Critical lengths of Steklov eigenvalues of hypersurfaces of revolution in Euclidean space

We study the Steklov problem on hypersurfaces of revolution with two boundary components in Euclidean space. In a recent article, the phenomenon of critical length, at which a Steklov eigenvalue is maximized, was exhibited and multiple questions were raised. In this article, we conjecture that, in any dimension, there is a finite number of infinite critical length. To investigate this, we develop an algorithm to efficiently perform numerical experiments, providing support to our conjecture. Furthermore, we prove the conjecture in dimension $n = 3$ and $n = 4$.

math.SP↗

arXiv2023

Sharp upper bounds for Steklov eigenvalues of a hypersurface of revolution with two boundary components in Euclidean space

We investigate the question of sharp upper bounds for the Steklov eigenvalues of a hypersurface of revolution of the Euclidean space with two boundary components isometric to two copies of $\mathbb{S}^{n-1}$. For the case of the first non zero Steklov eigenvalue, we give a sharp upper bound $B_n(L)$ (that depends only on the dimension $n \ge 3$ and the meridian length $L>0$) which is reached by a degenerated metric $g^*$, that we compute explicitly. We also give a sharp upper bound $B_n$ which depends only on $n$. Our method also permits us to prove some stability properties of these upper bounds.

math.DG↗

arXiv2023

Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs

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.

math.DG↗

arXiv2023

The Steklov problem on triangle-tiling graphs in the hyperbolic plane

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.

math.DG↗