I'm just trying to show that if $u\in W^{1,p}(U)$ (with $1\leq p<\infty$) then $|u|\in W^{1,p}(U)$. Here $U$ is bounded.
I originally took smooth functions $a_n$, equal to $|x|$ whenever $|x|\geq 1/n$ and which converged uniformly to $|x|$; this implies that $a_n(u)$ is Cauchy in $L^p$, so all I needed to do was show that $Da_n(u)$ was Cauchy, but then I ran into problems.
Namely, since no smooth functions approach the sign function uniformly, I can't seem to bound $||D_{x_i}(a_n(u)-a_m(u))||=||(a_n'(u)-a_m'(u))u_{x_i}||$.
Anyone know a way around this, or an easier approach? (If you're going to quote mollifiers, please be specific because I feel I've exhausted all approaches that use them!)
You don't need $a_n'$ to converge uniformly to the sign function; it's enough to have them uniformly bounded and converging pointwise. (If your chosen $a_n$ functions don't do this, then pick other ones that do.) Then you can use the dominated convergence theorem to show that $a_n'(u) u_{x_i} \to \operatorname{sgn}(u) u_{x_i}$ in $L^p$.