### Maximization of Neumann Eigenvalues

Analysis and Geometry Seminar

1st June 2022, 3:15 pm – 4:15 pm

Fry Building, Room 2.41

We discuss the maximization of the $k$-th eigenvalue of the Laplace operator with Neumann boundary conditions among domains of ${\mathbb R}^N$ with prescribed measure. We relax the problem to the class of (possibly degenerate) densities in ${\mathbb R}^N$ with prescribed mass and prove the existence of an optimal density. For $k=1,2$ the two problems are equivalent and the maximizers are known to be one and two equal balls, respectively. For $k \ge 3$ this question remains open, except in one dimension of the space where we prove that the maximal densities correspond to a union of $k$ equal segments. This result provides sharp upper bounds for Sturm-Liouville eigenvalues and proves the validity of the P\'olya conjecture in the class of densities in $\mathbb R$.

Based on the relaxed formulation, we provide numerical approximations of optimal densities for $k=1, \dots, 8$ in $\R^2$. This is a joint work with E. Martinet and E. Oudet.

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