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

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 t...

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Vydáno v:Journal of functional analysis Ročník 269; číslo 9; s. 2669 - 2708
Hlavní autoři: Balogh, Zoltán M., Calogero, Andrea, Kristály, Alexandru
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Inc 01.11.2015
Témata:
Aleksandrov-type maximum principle
Comparison principle
H-convex functions
Heisenberg group
Heisenberg group
Comparison principle
35R03
26B25
Aleksandrov-type maximum principle
H-convex functions
ISSN:0022-1236, 1096-0783
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Shrnutí: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.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2015.08.014