A free boundary approach to spectral optimization problems: Extremal Neumann eigenvalues

Abstract

This is an extended version of the invited talk given at the 9th European Congress of Mathematics. It is concerned with the geometric optimization of Neumann eigenvalues of the Laplace operator on domains of ${\mathbb{R}}^N$ and on spheres. The main question is to maximize the $k$-th eigenvalue of the Neumann Laplacian under a volume constraint; we approach it in the context of free boundary problems with no energy interface. Together with a discussion about existence/non-existence of optimal geometries and possible relaxation on densities, we point out how numerical approximations play a key role in understanding the problem. A particular attention is given to the lowest two non-trivial eigenvalues for which a full answer is available in any dimension of the Euclidean space and in the hyperbolic space, but not on spheres, where a surprising phenomenon occurs: while a complete answer can be given for the second eigenvalue, a series of open problems remain unsolved for the first one. The results presented in the talk have been obtained in collaborations with A. Henrot, R. Laugesen, E. Martinet, M. Nahon and E. Oudet.