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

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  • 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.
    Mathematics Subject Classification: Primary: 35B08; Secondary: 47G10.


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