Lax Formula for Obstacle Problems

The first order obstacle problem min{u(t) + H(Du), g(x) - u} = 0, u(T, x) = g(x) has a Hopf formula in the case when g is convex. It was first derived by A. Subbotin [11]. The case when g is continuous but the Hamiltonian H is convex is considered here. The corresponding Lax formula is derived to be u(t, x) = sup(y is an element of Rn) inf(t <=tau <= T) {g(y) - (tau - t)H* (y - x/tau - t)} = sup(y is an element of Rn) inf(t <=tau <= T) {g(x + y(tau - t)) - (tau - t)H* (y)}. This formula is shown to provide a viscosity solution of the obstacle problem. The argument to derive and prove this is based on optimal control in L-infinity.