Cauchy problems for noncoercive Hamilton–Jacobi–Isaacs equations with discontinuous coefficients

We study the Cauchy problem for a homogeneous and not necessarily coercive Hamilton-Jacobi-Isaacs equation with an x-dependent, piecewise continuous coefficient. We prove that under suitable assumptions there exists a unique and continuous viscosity solution. The result applies in particular to the Carnot-Caratheodory eikonal equation with discontinuous refraction index of a family of vector fields satisfying the Hormander condition. Our results are also of interest in connection with geometric flows with discontinuous velocity in anisotropic media with a non-euclidian ambient space