# A reference for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$

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.

• Could you provide any reference for $p=1,2$ ? – S. Maths Apr 6 at 14:47
• @S.Maths For $p=1$ you can find it in standard textbook like Evans' PDE or any book on Sobolev space. I've seen $p=2$ in some of the books as well and I believe someone wrote a proof on math.stackexchange.com. – BigbearZzz Apr 6 at 14:55