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

TL;DR

This paper establishes a sharp lower bound for the perimeter density at singular points of minimizing clusters in Euclidean space, providing a characterization of blow-ups at these points.

Contribution

It introduces a precise lower bound for perimeter density at singular points and proves the rigidity of this bound, characterizing blow-ups in minimizing clusters.

Findings

Established a sharp lower bound for perimeter density at singular points.

Proved that the lowest density characterizes the blow-up.

Provided a rigidity result for the structure of singularities.

Abstract

In this paper we study the blow-ups of the singular points in the boundary of a minimizing cluster lying in the interface of more than two chambers. We establish a sharp lower bound for the perimeter density at those points and we prove that this bound is rigid, namely having the lowest possible density completely characterizes the blow-up.