**Speaker**

Antoine Henrot

**Abstract**

Let $\mu_1(\Omega)$ be the first non-trivial eigenvalue of the Laplace operator with Neumann boundary conditions

on a smooth domain $\Omega$. It is a classical task to look for estimates of the eigenvalues involving geometric quantities like the area, the perimeter, the diameter…

In this talk, we will recall the classical inequalities known for $\mu_1$. Then we will focus on the following conjecture: prove that $P^2(\Omega) \mu_1(\Omega) \leq 16 \pi^2$ for all plane convex domains, the equality being achieved by the square AND the equilateral triangle. We will prove this conjecture assuming that $\Omega$ has two axis of symmetry.

This is a joint work with Antoine Lemenant and Ilaria Lucardesi