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

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.