The regularity of generalized solutions of Hamilton--Jacobi equations (Q1763689)

The Cauchy problem for Hamilton-Jacobi equations of the form \[ \begin{gathered} u_t+ H(t,\nabla_x u)= 0,\quad (t,x)\in\Omega= (0,T)\times \mathbb{R}^n,\\ u(0,x)= \sigma(x),\quad x\in \mathbb{R}^n,\end{gathered} \] is considered. By using the method of envelope, Hopf established two well-known formulas for the Lipschitz solutions of the kind (I) \(u(t,x)= \min_{y\in\mathbb{R}^n}\,\{\sigma(y)+ tH^*({x- y\over t})\}\) and (II) \(u(t,x)= \max_{q\in\mathbb{R}^n}\,\{(x, q)- \sigma^*(q)- tH(q)\}\). The author presents some results on the relationship between two Hopf's formulas and characteristics. A generalization of a result of Lions for formula (I) and another one for the formula (II), are established. Then, the domain where the generalized solutions given by Hopf's formulas are differentiable is studied.