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

In the Heisenberg group framework, we obtain a geometric inequality for stable solutions of \u394\u210d u = f (u) in a domain \u3a9 86\u210d. 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 \u3bd and the unit tangent by \u3c5, we have that Mathematic expression for any Mathematic expression. Stable solutions in the entire \u210d satisfying a suitably weighted energy growth and such that Mathematic expression are then shown to have level sets with vanishing mean curvature