A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems

Abstract

In the Heisenberg group framework, we obtain a geometric inequality for stable solutions of ${\Delta_{\mathbb{H}} u=f(u)}$ in a domain ${\Omega\subseteq\mathbb{H}}$ . More precisely, if we denote the horizontal intrinsic Hessian by Hu, the mean curvature of a level set by h, its imaginary curvature by p, the intrinsic normal by ν and the unit tangent by υ, we have that

$$\int \limits_{\Omega}|\, \nabla_{{{\mathbb{H}}}}\phi\, |^2|\, \nabla_{{{\mathbb{H}}}}u\, |^2\\ \quad\geq\int \limits_{\Omega \cap \{ \nabla_{{{\mathbb{H}}}}u\ne0\}} \left(|\, Hu\, |^2 -\langle(Hu)^2\nu,\nu\rangle_{{{\mathbb{H}}}} -2\left(TYuXu-TXuYu\right)\right)\phi^2\\ \\ \quad=\int \limits_{\Omega \cap\{ \nabla_{{{\mathbb{H}}}} u\ne0\}} |\, \nabla_{{{\mathbb{H}}}}u\, |^2 \Big[h^2+\Big(p+\frac{\langle Huv,\nu\rangle_{{{\mathbb{H}}}}}{|\, \nabla_{{{\mathbb{H}}}}u\, |} \Big)^2+2\langle T\nu,v \rangle_{{{\mathbb{H}}}}\Big]\phi^2$$

for any ${\phi\in C^\infty_0(\Omega)}$ . Stable solutions in the entire ${{\mathbb{H}}}$ satisfying a suitably weighted energy growth and such that ${\langle T\nu,v \rangle_{\mathbb{H}}\ge0}$ are then shown to have level sets with vanishing mean curvature.