Semiconcavity of the distance in the Heisenberg group

In this talk I present the existence of solution of Dirichlet problems for a class of fully non linear equations, acting in the Heisenberg group .The contest is that of viscosity solutions, since the operators we consider are of a non variational nature.It is well known how important is the role of the distance function for elliptic PDE in general. In the Heisenberg Group,we have the availability of two distances: the smooth distance and the Carnot-Carathéodory one.The Carnot-Carathéodory distance of a point from a compact set K is given bythe minimum time to reach K with "horizontal" curves of speed one. The horizontal curves are curves which are tangent to the space generating the Heisenberg algebra.In a recent preprint Cannarsa and Rifford proved that it is semiconcave using a very abstract proof. We shall use their result to build supersolutions in order to apply Perron method.