SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian

In this paper we consider a viscosity solution u of the Hamilton-Jacobi equation partial derivative(t)u + H(D(x)u) = 0 in Omega subset of [0,T] x R-n. where H is smooth and convex. We prove that when d(t,center dot) := H-p(D(x)u(t,center dot)), H-p := del H is BV for all t epsilon [0, T] and suitable hypotheses on the Lagrangian L hold, the Radon measure divd(t,center dot) can have Cantor part only for a countable number of t's in [0,T]. This result extends a result of Robyr for genuinely nonlinear scalar balance laws and a result of Bianchini, De Lellis and Robyr for uniformly convex Hamiltonians.