Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations
Qing Liu, Xiaodan Zhou · ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE · 2021
This paper introduces in a natural way a notion of horizontal convex envelopes of continuous functions in the Heisenberg group.We provide a convexification process to find the envelope in a constructive manner.We also apply the convexification process to show h-convexity of viscosity solutions to a class of fully nonlinear elliptic equations in the Heisenberg group satisfying a certain symmetry condition.Our examples show that in general one cannot expect h-convexity of solutions without the symmetry condition.