Viscosity solution of Hamilton Jacobi equation and pointwise differentiability a.e. Does the viscosity solution of the Hamilton Jacobi equation satisfy the equation pointwise a.e.?
 A: 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.
