Calculus of Variations and Geometric Measure Theory

V. Franceschi - A. Pratelli - G. Stefani

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

created by pratelli on 14 Jun 2021
modified by stefani on 16 May 2023


Published Paper

Inserted: 14 jun 2021
Last Updated: 16 may 2023

Journal: J. Éc. polytech. Math.
Volume: 10
Pages: 989--1045
Year: 2023

ArXiv: 2106.08099 PDF


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 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.