Marco Michetti

Optimal bounds for Neumann eigenvalues in terms of the diameter

In this talk we study the maximization problem of the Neumann eigenvalues under diameter constraint in an "optimal" class of domains. We define the profile function $f$ associated to a domain $\Omega\subset \mathbb{R}^d$ (defined as the $d-1$ dimensional measure of the slices orthogonal to a diameter), assuming that this function is $\beta$-concave we will give sharp upper bounds of the quantity $D(\Omega)^2\mu_k(\Omega)$ in terms of $\beta$. These optimal bounds will go to infinity when $\beta$ goes to zero giving in this way a geometric characterization of domains for which the diameter is fixed, but the Neumann eigenvalues are arbitrarily large. This includes the case of convex domains in $\mathbb{R}^d$, containing and generalizing previous results by P. Kröger. The proof of these results are based on a maximization problem for relaxed Sturm-Liouville eigenvalues.
This talk is based on a joint work with Antoine Henrot.