Rigidity results for non local phase transitions in the Heisenberg group $H$

TL;DR

This paper investigates the rigidity of stable solutions to fractional nonlocal equations in the Heisenberg group, establishing conditions under which their level sets are minimal surfaces with zero mean curvature.

Contribution

It introduces a Poincaré type inequality linked to a degenerate elliptic equation and uses an extension method to identify when solutions' level sets are minimal surfaces in the Heisenberg group.

Findings

Established a Poincaré type inequality for fractional equations in H

Derived a criterion for level sets to be minimal surfaces

Connected stability of solutions to geometric minimality in H

Abstract

In the Heisenberg group framework, we study rigidity properties for stable solutions of $(-\Delta_H)^s v = f(v)$ in $H$, $s \in (0,1)$. We obtain a Poincar\'e type inequality in connection with a degenerate elliptic equation in $\R^4_+$; through an extension (or "lifting") procedure, this inequality will be then used for giving a criterion under which the level sets of the above solutions are minimal surfaces in $H$, i.e. they have vanishing mean curvature.