Viscosity solutions of Hamilton--Jacobi equations with discontinuous coefficients

Giuseppe Maria Coclite, Nils Henrik Risebro

Published 2003-03-24Version 1

We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define a viscosity solution by treating the discontinuities in the coefficients analogously to ``internal boundaries''. By defining an appropriate penalization function, we prove that viscosity solutions are unique. The existence of viscosity solutions is established by showing that a sequence of front tracking approximations is compact in $L^\infty$, and that the limits are viscosity solutions.