Spectral Minimal Partitions: Local vs Global Minimality

In this overview talk we will explore a variational approach to the problem of Spectral Minimal Partitions (SMPs).  The problem is to partition a domain or a manifold into k subdomains so that the first Dirichlet eigenvalue on each subdomain is as small as possible.  We expand the problem to consider Spectral Critical Partitions (partitions where the max of the Dirichlet eigenvalues is experiencing a critical point) and show that a locally minimal bipartite partition is automatically globally minimal. Extensions of this result to non-bipartite partitions, as well as its connections to counting nodal domains of the eigenfunctions and to a two-sided Dirichlet-to-Neumann map defined on the partition boundaries, will also be discussed.

The talk is based on joint work with Yaiza Canzani, Graham Cox, Bernard Helffer, Peter Kuchment, Jeremy Marzuola, Uzy Smilansky and Mikael Sundqvist, with and without the speaker.