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.