Let $\Omega$ be an open domain with nice boundary and $u\in W^{1,p}(\Omega)$. I believe that $|u|^p\in W^{1,1}$ with $$ \nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u $$ but couldn't find a good reference for this except for the case $p=1,2$.
The assumption in standard textbooks that I usually see regarding the chain rule for Sobolev functions, i.e. $$ \nabla F(u) = F'(u)\nabla u, $$ is that $F\in C^1$ with bounded derivative (or perhaps a Lipschitz function). However, here $F(t)=|t|^p$, which doesn't satisfy that assumption.
Does anyone know where I can find this result in the literature?
I have a rough idea of how I would try to prove it myself, i.e. prove a similar result for $ F_\varepsilon(t) = (\varepsilon^2 + t^2)^{p/2} $ first then take $\varepsilon \to 0$. Still, I want to know where I can find a reference for this result.
