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

TL;DR

This paper extends the Steiner property to planar minimal clusters with anisotropic densities, showing they are composed of finitely many smooth arcs meeting at triple points under weak assumptions.

Contribution

It proves the Steiner property for anisotropic densities in the plane, generalizing previous isotropic results and allowing for direction-dependent perimeter densities.

Findings

Minimal clusters are made of finitely many $C^{1,eta}$ arcs.

Arcs meet at finitely many triple points.

The Steiner property holds under weak assumptions on densities.

Abstract

In this paper we discuss the Steiner property for minimal clusters in the plane with an anisotropic double density. This means that we consider the classical isoperimetric problem for clusters, but volume and perimeter are defined by using two densities. In particular, the perimeter density may also depend on the direction of the normal vector. The classical "Steiner property" for the Euclidean case (which corresponds to both densities being equal to $1$) says that minimal clusters are made by finitely many ${\rm C}^{1,\gamma}$ arcs, meeting in finitely many "triple points". We can show that this property holds under very weak assumptions on the densities. In the parallel paper "On the Steiner property for planar minimizing clusters. The isotropic case" we consider the isotropic case, i.e., when the perimeter density does not depend on the direction, which makes most of the construction much simpler. In particular, in the present case the three arcs at triple points do not necessarily meet with three angles of $120^\circ$, which is instead what happens in the isotropic case.