Calculus of Variations and Geometric Measure Theory

A. Cesaroni - M. Novaga

Periodic partitions with minimal perimeter

created by novaga on 22 Dec 2022
modified on 06 Mar 2024


Published Paper

Inserted: 22 dec 2022
Last Updated: 6 mar 2024

Journal: Nonlinear Analysis
Volume: 243
Number: Paper n. 113522
Pages: 16 pages
Year: 2024

ArXiv: 2212.11545 PDF


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.