Lattice tilings with minimal perimeter and unequal volumes

TL;DR

This paper investigates optimal periodic tessellations of Euclidean space with unequal cells, focusing on perimeter minimization, regularity, and the optimality of hexagonal tilings for nearly equal areas.

Contribution

It provides existence results, analyzes regularity under volume constraints, and proves hexagonal tilings are optimal for almost equal-area partitions.

Findings

Existence of perimeter-minimizing tessellations with unequal cells.

Regularity results under volume penalization and in planar cases.

Hexagonal tilings are optimal among partitions with nearly equal areas.

Abstract

We study periodic tessellations of the Euclidean space with unequal cells arising from the minimization of perimeter functionals. Existence results and qualitative properties of minimizers are discussed for different classes of problems, involving local and non-local perimeters. Regularity is then addressed in the general case under volume penalization, and in the planar case with the standard perimeter, prescribing the volumes of each cell. Finally, we show the optimality of hexagonal tilings among partitions with almost equal areas.