SBV Regularity for Hamilton-Jacobi Equations with Hamiltonian Depending on (t, x)

In this paper we prove the SBV regularity of the distributional derivative of a viscosity solution of the Hamilton-Jacobi equation ∂tu + H(t, x, Dxu) = 0 in Ω ⊂ [0, T ]× R, under the hypothesis of uniform convexity of the Hamiltonian H in the last variable. This result extends the result of Bianchini, De Lellis and Robyr obtained for an Hamiltonian H = H(Dxu) which depends only on the spatial distributional derivative of the solution.