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