Viscosity Solutions of Hamilton-Jacobi Equations with Discontinuous Coefficients

We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an $L^\infty$ contractive semigroup. We define a viscosity solution by treating the discontinuities in the coefficients analogously to ``internal boundaries''. The existence of viscosity solutions is established constructively via a front tracking approximation, whose limits are viscosity solutions, where by ``viscosity solution'' we mean a viscosity solution that posses some additional regularity at the discontinuities in the coefficients. We then show a comparison result that is valid for these viscosity solutions.