### Maximization of Neumann eigenvalues

Analysis and Geometry Seminar

20th February 2018, 3:00 pm – 3:55 pm

Main Maths Building, SM4

In this talk I will discuss the question of the maximization

of the $k$-th eigenvalue of the Neumann-Laplacian under a volume

constraint. After an introduction to the topic and discussion about the

existence of optimal geometries (and relaxation to densities), I will

focus on the low eigenvalues. The first non-trivial one is maximized by

the ball, the result being due to Szego and Weinberger in the fifties.

Concerning the second non-trivial eigenvalue, Girouard, Nadirashvili

and Polterovich proved that the supremum in the family of planar simply

connected domains of $R^2$ is attained by the union of two disjoint,

equal discs. I will show that a similar statement holds in any dimension

and without topological restrictions. This last result is jointly

obtained with A. Henrot.

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