Regularity results for the minimum time function with Hörmander vector fields

In a bounded domain of Rn with boundary given by a smooth (n−1)-dimensional manifold, we consider the homogeneous Dirichlet problem for the eikonal equation associated with a family of smooth vector fields {X1,…,XN} subject to Hörmander's bracket generating condition. We investigate the regularity of the viscosity solution T of such problem. Due to the presence of characteristic boundary points, singular trajectories may occur. First, we characterize these trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. Then, we prove that the local Lipschitz continuity of T, the local semiconcavity of T, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied whenever the characteristic set of {X1,…,XN} is a symplectic manifold. We apply our results to several examples.