Thomas Hoffmann-Ostenhof: Old and new results on spectral minimal partitions

We consider Dirichlet Laplacians on bounded domains mostly in R^2. Associated to such a domain we define for each positive integer k a spectral minimal partition. Such a partition is a specific, not necessarily unique, partition of the domain into k pairwise disjoint subsets which satisfy some natural minimizing spectral conditions. This is a nonlinear variational problem leading to partition eigenvalues L_k and the associated minimal partitions P_k. It turns out that there is an interesting connection between those minimal partitions P_k and nodal domains of eigenfunctions. Furthermore by analyzing spectral minimal partitions we achieve a characterization of the case when there is equality in Courant's nodal theorem. One of the main problems is to understand minimal partitions when they do not arise from eigenfunctions. We investigate specific examples and finally give a new characterization of those L_k, respectively P_k via specific eigenvalues and eigenfunctions of Aharonov Bohm Hamiltoninas.

This is joint work with Bernard Helffer, Susanna Terracini and partly with Virginie Bonnaillie-Noel.