Sobolev .Absolute.

Question

We have the result that : $\Omega$ is open set .If $u \in H^1(\Omega)$ then $|u|\in H^1(\Omega)$.My question is :

If $u \in H^2(\Omega) $ .Does $|u| \in H^2(\Omega)$?

Answer 1

No. Consider for example $\Omega = (-1,1)$ and $u(x) =x$. Then $w(x) = \vert u(x) \vert = \vert x \vert$ satisfies $w'(x) = 1$ for $x>0$ and $w'(x) =-1$ for $x \le 0$. If $w'$ were in $H^1((-1,1))$, it would be continuous, which it obviously isn't.