On viscosity solutions of the Hamilton-Jacobi equation (Q1962710)

The author investigates the value function \(V(t,x)\) of a Bolza problem, namely to minimize \[ \varphi(X(0)) + \int_0^t L(X(\tau),\dot{X}(\tau)) d\tau ,\quad X(t) = x . \] He proves that, under certain assumptions on the Lagrangian and on \(\varphi\), the value function is a viscosity subsolution of the associated Hamilton-Jacobi equation, and that it is locally Lipschitz continuous. He also proves some uniqueness and comparison results.