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 ∂ t u + H ( D x u ) = 0 in Ω ⊂ [ 0 , T ] × R n , where H is smooth and convex. We prove that when d ( t , ⋅ ) : = H p ( D x u ( t , ⋅ ) ) , H p : = ∇ H is BV for all t ∈ [ 0 , T ] and suitable hypotheses on the Lagrangian L hold, the Radon measure div d ( t , ⋅ ) can have Cantor part only for a countable number of tʼs 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.