Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

Abstract

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for $H-$convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous $H-$convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.

Published in: J. Funct. Anal. 269 (2015), no. 9, 2669-2708

References in corpus (3)

• Horizontal normal map on the Heisenberg group
• On the second order derivatives of convex functions on the Heisenberg group
• c horizontal convexity on Carnot groups

Cited by in corpus (3)

• Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations
• Hölder regularity of viscosity solutions of some fully nonlinear equations in the Heisenberg group
• The quaternionic Monge-Ampère operator and plurisubharmonic functions on the Heisenberg group