From pointwise to local regularity for solutions of Hamilton-Jacobi-Bellman equations

It is well-known that solutions to the Hamilton–Jacobi equation $u_t(t,x)+ H(x,ux(t,x)) = 0$ fail to be everywhere differentiable. Nevertheless, suppose a solution $u$ turns out to be dif- ferentiable at a given point $(t , x)$ in the interior of its domain. May then one deduce that $u$ must be continuously differentiable in a neighborhood of $(t, x)$? Although this question has a negative answer in general, our main result shows that it is indeed the case when the proximal subdifferential of $u(t, ·)$ at $x$ is nonempty. Our approach uses the representation of u as the value function of a Bolza problem in the calculus of variations, as well as necessary conditions for such a problem.