# American Institute of Mathematical Sciences

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June  2014, 34(6): 2639-2656. doi: 10.3934/dcds.2014.34.2639

## Rigidity results for nonlocal phase transitions in the Heisenberg group $\mathbb{H}$

 1 Departamento de Ingeniería Matemática, Universidad de Chile, Blanco Encalada 2120, Santiago, Chile 2 Laboratoire d'Analyse, Topologie, Probabilités, Aix-Marseille Université, 9, rue F. Joliot Curie, 13453 Marseille Cedex 13, France

Received  April 2013 Revised  May 2013 Published  December 2013

In the Heisenberg group framework, we study rigidity properties for stable solutions of $(-\Delta_\mathbb{H})^sv=f(v)$ in $\mathbb{H}$, $s\in(0,1)$. We obtain a Poincaré type inequality in connection with a degenerate elliptic equation in $\mathbb{R}^4_+$; 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 $\mathbb{H}$, i.e. they have vanishing mean curvature.
Citation: Luis F. López, Yannick Sire. Rigidity results for nonlocal phase transitions in the Heisenberg group $\mathbb{H}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2639-2656. doi: 10.3934/dcds.2014.34.2639
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