Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

2019/07/03 by Qing Liu, Liu, Qing, Xiaodan Zhou +1
Computer Science · Mathematics · #22E30 #26B25 #35D40 #35R03 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1907.01701

openalex publication_date 2019/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper introduces in a natural way a notion of horizontal convex envelopes of continuous functions in the Heisenberg group. We provide a convexification process to find the envelope in a constructive manner. We also apply the convexification process to show h-convexity of viscosity solutions to a class of fully nonlinear elliptic equations in the Heisenberg group satisfying a certain symmetry condition. Our examples show that in general one cannot expect h-convexity of solutions without the symmetry condition.

Related

• Horizontally quasiconvex envelope in the Heisenberg group
• Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group
• Horizontal normal map on the Heisenberg group
• Geodetically convex sets in the Heisenberg group \mathbb Hⁿ
• The engulfing property for sections of convex functions in the Heisenberg group and the associated quasi--metric