Lattice tilings minimizing nonlocal perimeters

TL;DR

This paper establishes the existence of periodic lattice tilings that minimize a broad class of nonlocal perimeter functionals, extending classical isoperimetric problems to nonlocal interactions and discussing optimality in the plane.

Contribution

It proves the existence of minimizers for nonlocal perimeter functionals in periodic tessellations and reformulates the problem as an isoperimetric problem among fundamental domains.

Findings

Existence of periodic tessellations minimizing nonlocal perimeters.

Reformulation of the problem as an isoperimetric problem among lattice fundamental domains.

Discussion on the potential optimality of hexagonal tilings in the plane.

Abstract

We prove the existence of periodic tessellations of $\mathbb{R}^N$ minimizing a general nonlocal perimeter functional, defined as the interaction between a set and its complement through a nonnegative kernel, which we assume to be either integrable at the origin, or singular, with a fractional type singularity. We reformulate the optimal partition problem as an isoperimetric problem among fundamental domains associated with discrete subgroups of $\mathbb{R}^N$ , and we provide the existence of a solution by using suitable concentrated compactness type arguments and compactness results for lattices. Finally, we discuss the possible optimality of the hexagonal tessellation in the planar case.