Mathematics

A cluster is a partition of $\mathbb{R}^n$ into a finite family of sets of finite perimeter called chambers. We shall study the singular points of minimizing clusters (i.e. partitions minimizing the perimeter under some volume constraints); in particular we shall investigate the peculiar behavior of the singular points lying in the boundary of three or more chambers. In the talk we will establish a dimension-free lower bound for the perimeter density at those "special" singular points and we will discuss some applications. The results are obtained in collaboration with Jonas Hirsch.