Workshop: Bounds for the first (non-trivial) Neumann eigenvalue and partial results on a nice conjecture

Date: 2022-11-01

Time: 10:00 - 11:00

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Antoine Henrot


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