Ilias Ftouhi – Mathematics and Decision

$\begin{quote} \begin{center} \textbf{Shape optimization and estimation of the fundamental frequency} \end{center} \medskip Spectral theory has been interesting various communities in the last century and still is. We are particularly 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: Perimeter $P(\Omega)$ and measure $|\Omega|$). 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-Santalo diagram of the triplet $(\lambda_1,P,|\cdot|)$: $$\mathcal{C}_{\mathcal{F}_{ad}}:=\big\{ \big(P(\Omega),\lambda_1(\Omega)\big)\ |\ \Omega \in \mathcal{F}_{ad}\ \text{and}\ |\Omega|=1 \big\},$$ where $\mathcal{F}_{ad}$ is a given class of subsets of $\mathbb{R}^d$. Notice that the characterization of such diagram is equivalent to finding all the possible scale-invariant inequalities between the three involved quantities ($P$, $\lambda_1$ and $|.|$ in our case). We are able to completely describe the diagram for open sets in $\mathbb{R}^d$. It shows that there is no other inequality than the Faber-Krahn and the isoperimetric ones. This motivates us to investigate what happens for other classes of sets like the simply-connected and convex ones. We will give an advanced description of the Blaschke-Santalo diagram for convex sets in the plane. This work is in collaboration with Jimmy Lamboley. \end{quote}$