In the Heisenberg group framework, we obtain a geometric inequality for stable solutions of Δℍ u = f (u) in a domain Ω⊆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 Mathematic expression for any Mathematic expression. Stable solutions in the entire ℍ satisfying a suitably weighted energy growth and such that Mathematic expression are then shown to have level sets with vanishing mean curvature.
A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems / F. Ferrari, E. Valdinoci. - In: MATHEMATISCHE ANNALEN. - ISSN 0025-5831. - 343:2(2009), pp. 351-370.
A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems
E. ValdinociPrimo
2009
Abstract
In the Heisenberg group framework, we obtain a geometric inequality for stable solutions of Δℍ u = f (u) in a domain Ω⊆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 Mathematic expression for any Mathematic expression. Stable solutions in the entire ℍ satisfying a suitably weighted energy growth and such that Mathematic expression are then shown to have level sets with vanishing mean curvature.Pubblicazioni consigliate
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