Minimal periodic foams with fixed inradius

TL;DR

This paper proves the existence and regularity of periodic tilings of Euclidean space into equal cells with a fixed inradius that minimize classical or fractional perimeter, exploring properties in dimensions 3 and 4.

Contribution

It establishes the existence, regularity, and qualitative properties of minimal periodic foams with fixed inradius for both classical and fractional perimeters.

Findings

Existence of minimal periodic tilings with fixed inradius.

Regularity results for these tilings.

Qualitative properties in dimensions 3 and 4.

Abstract

In this note we show existence and regularity of periodic tilings of the Euclidean space into equal cells containing a ball of fixed radius, which minimize either the classical or the fractional perimeter. We also discuss some qualitative properties of minimizers in dimensions $3$ and $4$.