SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian
Résumé
In this paper we consider a viscosity solution u of the Hamilton-Jacobi equation where H is smooth and convex. We prove that when d(t,⋅):=Hp(Dxu(t,⋅)), Hp:=∇H is BV for all t∈[0,T] and suitable hypotheses on the Lagrangian L hold, the Radon measure 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.