Existence theorems for fully nonlinear equations in the Heisenberg Group

The principal aim of this work is to prove 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-Carathéeodory distance from a set K is given by the minimum time to reach K from a point 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 will use this result to prove the existence theorem.