A version of the Hopf-Lax formula in the Heisenberg group

We consider Hamilton-Jacobi equations u(t) + H(D(h)u) = 0 in the H x R+, where H is the Heisenberg group and D(h)u denotes the horizontal gradient of u. We establish uniqueness of bounded Viscosity Solutions with continuous initial data u(p, 0) = g(p). When the hamiltonian H is radial, convex and superlinear the Solution is given by the Hopf-Lax formula u(p,t) = inf(qepsilonH){tL(q-1.p/t) + g(q), where the Lagrangian L is the horizontal Legendre transform of H lifted to H by requiring it to be radial with respect to the Carnot-Caratheodory metric.