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

It is well-known that solutions to the Hamilton-Jacobi equation 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 $$u(t,\cdot )$$ 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. [ABSTRACT FROM AUTHOR]