Formula for a solution of u_t + H(u,Du) = g

Abstract

We study the continuous as well as the discontinuous solutions of Hamilton-Jacobi equation u_t + H(u,Du) = g in Rⁿ × R₊ with u(x, 0) = u₀(x). The Hamiltonian H(s,p) is assumed to be convex and positively homogeneous of degree one in p for each s in R. If H is non increasing in s, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous viscosity solutions.