Suppose $\Omega \subset \mathbb{R}^3$ is bounded domain with smooth boundary and $u \in W^{1,2}(\Omega,\mathbb{S}^2)$ such that $|u|=1$ a.e. in $\Omega$. It is known that there is a sequence $u_n \in C^1(\overline{\Omega})$ such that $u_n \to u$ in $W^{1,2}$ norm.
Is it possible to find such a sequence with $|u_n|=1$ for all $n$ in $\Omega$ (or just $|u_n|=1$ on $\partial \Omega$)?
I found that the construction of such sequence is the use of mollifier but I can't figure out how to make $u_n$ satisfies $|u_n|=1$.
Also, I have tried to show that $u_n/|u_n|$ converges to $u$ in $W^{1,2}$ norm, but not success. Thank you!