Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients (Q605504)

Summary: 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-Carathéodory eikonal equation with discontinuous refraction index of a family of vector fields satisfying the Hörmander condition. Our results are also of interest in connection with geometric flows with discontinuous velocity in anisotropic media with a non-Euclidian ambient space.