On the estimation of the fundamental frequency (Ilias Ftouhi, FAU Erlangen-Nürnberg)

Séminaire « Analyse numérique et équations aux dérivées partielles »
En visio
We are interested in the spectrum of the Laplace operator with Dirichlet boundary conditions on $\partial \Omega$ where $\Omega\subset \mathbb{R}^d$, and more precisely its first eigenvalue also known as the fundamental frequency $\lambda_1(\Omega)$. Unfortunately, for almost all given sets $\Omega$, there is no explicit formula for $\lambda_1(\Omega)$. This motivates to look for estimates via other functionals, which are much easier to compute (for example: the perimeter $P(\Omega)$ and the volume $|\Omega|$). First, we give a brief introduction on shape optimization and spectral theory, then we introduce the following set of points, which can be called the Blaschke--Santal\'o diagram of the triplet $(P,\lambda_1,|\cdot|)$: $$C_{F_{ad}} := \{(P(\Omega),\lambda_1(\Omega))\ |\ \Omega\in F_{ad}\ \text{and}\ |\Omega|=1\},$$
where $F_{ad}$ is a given class of subsets of $\mathbb{R}^d$. Notice that the characterization of such diagram is equivalent to finding all the possible inequalities between the three involved quantities ($P$, $\lambda_1$ and $|\cdot|$ in our case). We are able to completely describe the diagram for open sets in $\mathbb{R}^d$ showing that there is no other inequality than the Faber--Krahn and the isoperimetric ones. This motivates to investigate what happens for other classes of sets like convex ones, for which we provide an advanced description of the corresponding Blaschke--Santal\'o diagram. This work is in collaboration with Jimmy Lamboley.

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