SBV regularity for Hamilton-Jacobi equations in ${\mathbb{R}}^n$

In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $${\mathrm{\partial }}_{t}u+H(D_{x}u)=0\text{in}\mathrm{\Omega }\subset \mathbb{R}\times {\mathbb{R}}^n\mathrm{.}$$ In particular, under the assumption that the Hamiltonian $H\in C^2({\mathbb{R}}^n)$ is uniformly convex, we prove that $D_{x}u$ and ${\mathrm{\partial }}_{t}u$ belong to the class $SB{V}_{loc}(\mathrm{\Omega })$.