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∂tu+H(Dxu)=0in Ω⊂[0,T]×Rn, where H is smooth and convex. We prove that when d(t, {dot operator}):=H p(D xu(t, {dot operator})), H p:=∇;H is BV for all t∈[0, T] and suitable hypotheses on the Lagrangian L hold, the Radon measure divd(t, {dot operator}) 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. © 2012 Elsevier Inc.