Viscosity solution of Hamilton Jacobi equation and pointwise differentiability a.e.

Question

Does the viscosity solution of the Hamilton Jacobi equation satisfy the equation pointwise a.e.?

Answer 1

1) A viscosity solution satisfies pointwise the equation at every point of differentiability.

Namely, let $u\in C(\Omega)$ be a viscosity solution of $$ F(x, u(x), Du(x)) = 0\qquad \text{in}\ \Omega, $$ i.e., for every $x\in\Omega$, $$ (1)\quad F(x, u(x), p) \leq 0 \quad \forall p\in D^+ u(x), \qquad F(x, u(x), p) \geq 0 \quad \forall p\in D^- u(x). $$ Assume that $u$ is differentiable at some point $x_0\in\Omega$. This means that $$ D^+ u(x_0) = D^- u(x_0) = \{\nabla u (x_0)\}, $$ hence, using (1), $$ 0\leq F(x_0, u(x_0), \nabla u(x_0)) \leq 0. $$

EDIT: proof with test functions. If $u$ is differentiable at $x_0$, then $\nabla\varphi(x_0) = \nabla u(x_0)$ for every test function $\varphi$ touching $u$ at $x_0$ from above or from below. Hence $$ 0\leq F(x_0, u(x_0), \nabla u(x_0)) \leq 0. $$

2) For many HJ equations, solutions are differentiable a.e.

For example, if you consider an HJ equation of the form $$ u + H(x, Du(x)) = 0 $$ with $H(x, p) \to +\infty$ as $|p|\to +\infty$, then any bounded and continuous solution $u$ is also locally Lipschitz continuous (so that, by Rademacher's theorem, it is differentiable a.e.). See e.g. Bardi-Capuzzo Dolcetta, "Optimal Control...", Section 4.1.