Optimal control with discontinuous running cost: Eikonal equation and shape-from-shading

We consider a class of Hamilton-Jacobi equations with discontinuous coefficients that contains as examples the eikonal equation with discontinuous refraction index, or the Hamilton-Jacobi equation of shape-from-shading with discontinuous intensity function. In general the Dirichlet problem for such equations does not have unique solution, however we can characterize the minimal and maximal solution and, under appropriate compatibility conditions, prove existence of a unique solution in the sense of viscosity solutions