MIT PDE/Analysis Seminar

Tuesday, October 31, 2023 at 4:00pm to 5:00pm

MIT, 4-231 or 2-136 182 Memorial Drive, Cambridge, MA

Speaker:   Gregory Berkolaiko (Texas A&M)

Title:  Spectral minimal partitions: local vs global minimality

Abstract:  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 several papers of Yaiza Canzani, Graham Cox,
Bernard Helffer, Peter Kuchment, Jeremy Marzuola, Uzy Smilansky and
Mikael Sundqvist, with and without the speaker.


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