Upper bounds for Steklov eigenvalues : from graphs and discretization to hypersurfaces of revolution and numerical experiments
Léonard Tschanz · 2023
One aim of spectral geometry is to understand the relationship between the geometry or topol- ogy of a Riemannian manifold and the spectrum of a differential Laplacian-type operator asso- ciated to that manifold. This concept is encapsulated in the famous sentence of Mark Kac "Can one hear the shape of a drum?" and constitutes an active topic of fundamental research. In this thesis we consider the Steklov problem. In a first part of the dissertation, we find upper bounds for the Steklov eigenvalues σk(Ω, B), where (Ω, B) is a subgraph of a host graph Γ. The procedure used here consists in building a manifold M from the subgraph in such a way that we control σk(M), and transfer the spectral piece of information to the subgraph thanks to a method called discretization. This procedure allows us to work on two different classes of host graphs: a first class consists of Cayley graphs of polynomial growth groups, and a second class consists of triangle-tiling graphs of the hyperbolic plane. In a second part of the dissertation, we find sharp upper bounds for the first Steklov eigenvalue of a hypersurface of revolution M with two boundary components of the Euclidean space. To do that, we compare σ1(M) with σD 0 (A) and σN 1 (A), the first non trivial eigenvalues of the mixed Steklov-Dirichlet and Steklov-Neumann problems on a Euclidean annulus A. To extend the result to every eigenvalue, we introduce the concept of finite and infinite critical lengths, which makes us perform some numerical experiments that support a conjecture presented in the last chapter.