Hopf-Lax formulas for semicontinuous data

The equations u(t) + H(Du) = 0 and u(t) + H(u, Du) = 0, with initial condit ion u(0,x) = g(x) have an explicit solution when the hamiltonian is convex in the gradient variable (Lax formula) or the initial data is convex, of qu asiconvex (Hopf formula). This paper extends these formulas to initial func tions g which are only lower semicontinuous (lsc), and possibly infinite. I t is proved that the Lax formulas give a Ise viscosity solution, and the Ho pf formulas result in the minimal supersolution. A level set approach is us ed to give the most general results.