Periodic partitions with minimal perimeter

TL;DR

This paper proves the existence and regularity of minimal perimeter partitions in various metric measure spaces, extending known results and providing detailed descriptions in the planar case.

Contribution

It establishes existence and regularity of minimal perimeter partitions in general spaces, including fractional and anisotropic perimeters, with new results on local finiteness and detailed planar classifications.

Findings

Existence of minimal perimeter fundamental domains in homogeneous metric measure spaces.

Regularity results for minimal domains in specific cases like fractional perimeter.

In the plane, a detailed classification of minimal anisotropic perimeter domains.

Abstract

We show existence of fundamental domains which minimize a general perimeter functional in a homogeneous metric measure space. In some cases, which include the usual perimeter in the universal cover of a closed Riemannian manifold, and the fractional perimeter in $\mathbb R^n$, we can prove regularity of the minimal domains. As a byproduct of our analysis we obtain that a countable partition which is minimal for the fractional perimeter is locally finite and regular, extending a result previously known for the local perimeter. Finally, in the planar case we provide a detailed description of the fundamental domains which are minimal for a general anisotropic perimeter.