Hopf-Lax Formulas for Semicontinuous Data

Abstract

The equations ut + H(Du) = 0 and ut + H(u, Du) = 0, with initial condition 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, or quasiconvex (Hopf formula). This paper extends these formulas to initial functions g which are only lower semicontinuous (lsc), and possibly infinite. It is proved that the Lax formulas give a lsc viscosity solution, and the Hopf formulas result in the minimal supersolution. A level set approach is used to give the most general results.