Maximization of Neumann eigenvalues under diameter constraint

par Marco Michetti (LMO, Université Paris-Saclay)

lundi 28 nov. 2022, 10:00 → 11:00 Europe/Paris

Université Paris-Dauphine

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 $\mathrm{\Omega }\subset {\mathbb{R}}^d$, assuming that this function is $\beta$-concave, with $0<\beta \le 1$, we will give sharp upper bounds of the quantity $D(\mathrm{\Omega }{)}^2{\mu }_k(\mathrm{\Omega })$ in terms of $\beta$. These 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 eigenvalue are large. This will also give a new proof of a result by Kröger, namely sharp upper bounds for $D(\mathrm{\Omega }{)}^2{\mu }_k(\mathrm{\Omega })$ when $\mathrm{\Omega }$ is convex (that correspond to $\beta =(d-1{)}^{-1}$). The proof of this results are based on a maximization problem for relaxed Sturm-Liouville eigenvalues. This talk is based on a joint work with Antoine Henrot.