The regularity of generalized solutions of Hamilton--Jacobi equations
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.
scientific article; zbMATH DE number 2136553