On the shape of minimizers for the periodic nonlocal perimeter in $\mathbb{R}^2$

TL;DR

This paper investigates the shape and stability of periodic nonlocal minimal surfaces in two dimensions, providing evidence that large-area minimizers are straight bands, similar to classical local isoperimetric solutions.

Contribution

It proves that nonlocal Delaunay sets that are not straight bands are unstable, supporting the conjecture that large-area minimizers are straight bands in the nonlocal isoperimetric problem.

Findings

Nonlocal Delaunay sets not being straight bands are unstable.

Large-area minimizers are conjectured to be straight bands.

Results align with classical local isoperimetric solutions.

Abstract

In this paper, we study planar nonlocal Delaunay sets. That is, open sets in $\mathbb{R}^2$ with constant nonlocal mean curvature that are periodic in $x_1$, and even in $x_1$ and in $x_2$. Using bifurcation analysis and fine explicit computations, we prove that every sufficiently $C^{1,\beta}$-flat nonlocal Delaunay set in $\mathbb{R}^2$ that is not a straight band is unstable with respect to volume-preserving periodic variations. Our results support the conjecture that, as in the local case, in the range of large areas, minimizers of the periodic nonlocal isoperimetric problem -- also known as the nonlocal liquid drop problem with prescribed area between two parallel hyperplanes -- are all straight bands.