On the Eikonal equation for degenerate elliptic operators

We consider the nonnegative viscosity solution of the homogeneous Dirichlet problem for an eikonal equation associated to an operator sum of squares of vector fields of Grushin type in a symmetric domain. We show that the solution is locally Lipschitz continuous except at the characteristic boundary point. In the characteristic boundary point the solution has a Hölder regularity with exponent related to the Hörmander bracket condition. Finally, the singular set is an analytic stratification given by the characteristic boundary point and a half line