### Optimal bounds for Neumann eigenvalues in terms of the diameter

Analysis and Geometry Seminar

23rd February 2023, 3:30 pm – 3:30 pm

Fry Building, Room 2.04

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.

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