A. Calogero

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

In this paper we solve a problem raised by Gutierrez 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.

Optimization in the Heisenberg group

In this article, the local unconstrained and the constrained optimization problems in the Heisenberg group are investigated. The framework on which we work is given by the class of weakly H-convex functions recently introduced in the literature. This geometric notion of convexity, that is strictly related to the stratified structure of the group and has not an analogous in a Euclidean setting, requires a condition of convexity to be satisfied at the points of the horizontal planes. We find second-order sufficient conditions for a local extremum in both the unconstrained and the constrained optimization problems exploiting the weak H-convexity and the geometric behaviour of the horizontal planes in .