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

Abstract

In the Heisenberg group framework, we study rigidity properties for stable solutions of $(-Δ_H)^s v = f(v)$ in $H$, $s \in (0,1)$. We obtain a Poincaré 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.

Accepted in Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 2013. v2 quits some unnecessary properties and definitions. arXiv admin note: text overlap with arXiv:1206.0885 by other authors