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 fir st order, nonlinear equations containing the minimum time problem, who se solution is expected to be discontinuous. We prove existence, uniqu eness and representation formulas for the solution in the sense of vis cosity solutions. Our method relies on a new way of prescribing the bo undary condition, the use of recent ideas of Barron-Jensen [8] and Bar les [5], and the derivation of a ''backwards'' dynamic programming pri nciple. We use the same ideas to prove uniqueness for the usual Dirich let type formulation, following Ishii [13] and Barles-Perthame [6], un der additional regularity conditions on the domain.