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

摘要

In the Heisenberg group framework, we obtain a geometric inequality for stable solutions of ΔHu = f (u) in a domain ΩH. 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 v, we have that for any φ ∈ C_0~∞ (Ω). Stable solutions in the entire H satisfying a suitably weighted energy growth and such that (Tv, v)H ≥ 0 are then shown to have level sets with vanishing mean curvature.