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

2015; Elsevier BV; Volume: 269; Issue: 9 Linguagem: Inglês

10.1016/j.jfa.2015.08.014

1096-0783

Zoltán M. Balogh, Andrea Calogero, Alexandru Kristály,

Numerical methods in inverse problems

In this paper we solve a problem raised by Guti\'errez 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.