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

Abstract

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,⋅):=Hp(Dxu(t,⋅)), Hp:=∇H is BV for all t∈[0,T] and suitable hypotheses on the Lagrangian L hold, the Radon measure divd(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.