DISCONTINUOUS VISCOSITY SOLUTIONS TO DIRICHLET PROBLEMS FOR HAMILTON-JACOBI EQUATIONS WITH CONVEX HAMILTONIANS

We introduce a new formulation of Dirichlet problem for a class of first order, nonlinear equations containing the minimum time problem, whose solution is expected to be discontinuous. We prove existence, uniqueness and representation formulas for the solution in the sense of viscosity solutions. Our method relies on a new way of prescribing the boundary condition, the use of recent ideas of Barron-Jensen [8] and Barles [5], and the derivation of a ''backwards'' dynamic programming principle. We use the same ideas to prove uniqueness for the usual Dirichlet type formulation, following Ishii [13] and Barles-Perthame [6], under additional regularity conditions on the domain.