On the Steiner property for planar minimizing clusters. The isotropic case

TL;DR

This paper proves that in the isotropic case of the planar isoperimetric problem with double density, minimal clusters have boundaries composed of smooth arcs meeting at triple points with 120-degree angles.

Contribution

It establishes the Steiner property for minimal clusters in the isotropic double density setting, extending understanding of their regularity and structure.

Findings

Boundaries are ${ m C}^{1,eta}$ regular arcs.

Triple points occur with 120° angles.

Results hold in a wide generality.

Abstract

We consider the isoperimetric problem for clusters in the plane with a double density, that is, perimeter and volume depend on two weights. In this paper we consider the isotropic case, in the parallel paper "On the Steiner property for planar minimizing clusters. The anisotropic case", the anisotropic case is studied. Here we prove that, in a wide generality, minimal clusters enjoy the "Steiner property", which means that the boundaries are made by ${\rm C}^{1,\gamma}$ regular arcs, meeting in finitely many triple points with the $120^\circ$ property.