Solution of the fractional Allen-Cahn equation which are invariant under screw motion

TL;DR

This paper investigates entire solutions to the fractional Allen-Cahn equation in three-dimensional space that are invariant under screw motion and vanish on helicoids, also showing helicoids have zero nonlocal mean curvature.

Contribution

It establishes existence and non-existence results for solutions with specific symmetry and geometric properties, linking fractional PDEs with minimal surface theory.

Findings

Helicoids are surfaces with zero nonlocal mean curvature.

Existence and non-existence results for solutions vanish on helicoids.

Solutions are invariant under screw-motion symmetry.

Abstract

We establish existence and non-existence results for entire solutions to the fractional Allen-Cahn equation in $\mathbb R^3$, which vanish on helicoids and are invariant under screw-motion. In addition, we prove that helicoids are surfaces with vanishing nonlocal mean curvature.