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

Abstract

International audienceIn this paper we consider a viscosity solution u of the Hamilton-Jacobi equation where the Hamiltonian H is smooth and convex. We prove that when the vector field d(t, ):=H_pD_xu(t, )), is BV for all t in[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 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