Some regularity properties of viscosity solution defined by Hopf formula

Some properties of characteristic curves in connection with viscosity solution of Hamilton-Jacobi equations $(H,\sigma)$ defined by Hopf formula $u(t,x)=\max_{q\in\R^n}\{ \langle x,q\rangle -\sigma^*(q)-tH(q)\}$ are studied. We are concerned with the points where the solution $u(t,x)$ is differentiable, and the strip of the form $\mathcal R=(0,t_0)\times \R^n$ of the domain $\Omega$ where $u(t,x)$ is of class $C^1(\mathcal R).$ Moreover, we investigate the propagation of singularities in forward of this solution.