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 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.

Regulating Water Footprint in Trade: The Role of WTO Rules & Principles

World Water Week is hosted and organised by the Stockholm International Water Institute (SIWI) and takes place each year in Stockholm. The World Water Week has been the annual focal point for the globe's water issues since 1991. Every year, over 200 collaborating organisations convene events at the World Water Week. In addition, individuals from around the globe present their findings at the scientific workshops.