Geometric Analysis and Topology Seminar

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Lower bound for the perimeter density at singular points of a minimizing cluster in R^N

Speaker: Jonas Hirsch, SISSA

Location: Warren Weaver Hall 517

Date: Wednesday, March 21, 2018, 11 a.m.

Synopsis:

In this seminar we present results about the blow-ups of the singular points in the boundary of a minimizing cluster lying in the interface of more than two chambers.

To be more detailed, we deal with the study of the singularities of an isoperimetric cluster, that is a finite and disjoint family of sets of finite perimeter, called chambers, which minimizes the sum of the perimeter of all the chambers. Existence and regularity in the reduced boundary of the cluster is nowadays classical. Whereas the description of the singular part is almost completely open. 

Firstly let us note that a single minimal hyper-surface in dimension N leq 8 does not develop any singularities. In contrast any point in the interface of three or more different chambers is a singular point of minimizing cluster. Only in dimensions N leq 3 there is a complete characterization due to the seminal work by J.Taylor. 

The aim of this seminar will be to present that by induction on the dimension one can nonetheless obtain a sharp lower bound:

 

Theta_0(E) geq 3/2.

 

Even more, the density 3/2 completely characterizes the blow-up.