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

Franceschi, Valentina; Pratelli, Aldo; Stefani, Giorgio

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 989-1045

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 ${\mathrm{C}}^{1,\gamma }$ 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 [13] 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.

Dans cet article, nous discutons de la propriété de Steiner pour les clusters minimaux dans le plan avec une double densité anisotrope. Cela signifie que nous considérons le problème isopérimétrique classique pour les clusters, mais que le volume et le périmètre sont définis à l’aide de deux densités. En particulier, la densité du périmètre peut également dépendre de la direction du vecteur normal. La « propriété de Steiner » classique pour le cas euclidien (qui correspond aux deux densités égales à $1$) dit que les clusters minimaux sont constitués d’un nombre fini d’arcs ${\mathrm{C}}^{1,\gamma }$, se rencontrant en un nombre fini de « points triples ». Nous pouvons montrer que cette propriété est valable sous des hypothèses très faibles sur les densités. Dans l’article parallèle [13], nous considérons le cas isotrope, c’est-à-dire lorsque la densité du périmètre ne dépend pas de la direction, ce qui rend la plupart des constructions beaucoup plus simples. En particulier, dans le cas présent, les trois arcs aux points triples ne se rencontrent pas nécessairement avec trois angles de ${120}^{\circ }$, contrairement à ce qui arrive dans le cas isotrope.

Classification : 49Q10, 49Q20
Keywords: Perimeter and volume with density, clustering isoperimetric problem, Steiner property, anisotropic perimeter
Mots-clés : Densités du périmètre et du volume, problème isopérimétrique pour les clusters, propriété de Steiner, périmètre anisotrope

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