  We prove optimal regularity for the double obstacle problem when obstacles
are given by solutions to Hamilton-Jacobi equations that are not $C^2$.
  When the Hamilton-Jacobi equation is not $C^2$ then the standard Bernstein
technique fails and we loose the usual semi-concavity estimates. Using a
non-homogeneous scaling (different speed in different directions) we develop a
new pointwise regularity theory for Hamilton-Jacobi equations at points where
the solution touches the obstacle.
  A consequence of our result is that $C^1$-solutions to the Hamilton-Jacobi
equation $$ \pm |\nabla h-a(x)|^2=\pm 1 \textrm{in} B_1, \qquad h=f \textrm{on}
\partial B_1, $$ are in fact $C^{1,\alpha/2}$ provided that $a \in C^\alpha$.
This result is optimal and to the authors' best knowledge new.
