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.

  • $\begingroup$ Could you provide any reference for $p=1,2$ ? $\endgroup$ – S. Maths Apr 6 at 14:47
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    $\begingroup$ @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. $\endgroup$ – BigbearZzz Apr 6 at 14:55

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