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

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 ${\mathrm{C}}^{1,\gamma }$ regular arcs, meeting in finitely many triple points with the $120^{\circ }$ property.

arXiv commentScheme of the proof and references added. arXiv admin note: text overlap with arXiv:2106.08099

References in corpus (7)

• Proof of the Double Bubble Conjecture in R^n
• On the isoperimetric problem with double density
• The $\epsilon -{\epsilon }^{\beta }$ property in the isoperimetric problem with double density, and the regularity of isoperimetric sets
• Isoperimetric problem in H-type groups and Grushin spaces
• Minimal clusters of four planar regions with the same area
• Symmetric double bubbles in the Grushin plane
• A minimal partition problem with trace constraint in the Grushin plane

Cited by in corpus (3)

• On the Steiner property for planar minimizing clusters. The anisotropic case
• On the $N$-Cheeger problem for component-wise increasing norms
• On the monotonicity of weighted perimeters of convex bodies