Geometric second derivative estimates in Carnot groups and convexity

2008/03/07 by Nicola Garofalo, Garofalo, Nicola
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.0803.1021

openalex publication_date 2008/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove some new a priori estimates for H₂-convex functions which are zero on the boundary of a bounded smooth domain Ωin a Carnot group G. Such estimates are global and are geometric in nature as they involve the horizontal mean curvature \mathcal H of the boundary of Ω. As a consequence of our bounds we show that if G has step two, then for any smooth H₂-convex function in Ω⊂ G vanishing on the boundary of Ωone has ∑_i,j=1^m ∫_Ω([X_i,X_j]u)² dg ≤ 4/3 ∫_{∂ Ω} \mathcal H |∇_H u|² dσ_H .

Citations

• Constant mean curvature surfaces in sub-Riemannian geometry

Related

• c horizontal convexity on Carnot groups
• Viscosity convex functions on Carnot groups
• Isoperimetric, Sobolev and Poincaré inequalities on hypersurfaces in sub-Riemannian Carnot groups
• Geometric inequalities in Carnot groups
• Isoperimetric and Sobolev inequalities on hypersurfaces in sub-Riemannian Carnot groups