From pointwise to local regularity for solutions of Hamilton–Jacobi equations

It is well-known that solutions to the Hamilton-Jacobi equation ut(t, x) + H(x,ux(t, x)) = 0 fail to be everywhere differentiable. Nevertheless, suppose a solution u turns out to be differentiable 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.