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

Stefano Bianchini (1) , Daniela Tonon (2)

1 SISSA / ISAS - Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies
2 IMJ - Institut de Mathématiques de Jussieu

In 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.

Domaines: Equations aux dérivées partielles [math.AP]

Stefano Bianchini, Daniela Tonon. SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian. Journal of Mathematical Analysis and Applications, 2012, 391, pp.190-208. ⟨hal-00918446⟩