Let $\Omega \subset \mathbb{R}^{N}$ and $W^{1,p}(\Omega)$ be Sobolev Space. Then we let $u\in W^{1,p}(\Omega)$ and define
(i) $u^{+} := \max\{0,u\}$
(ii) $u^{-} := \max\{0,-u\}$
Then, we claim that $u^{+}, u^{-}\in W^{1,p}(\Omega)$ and we have
\begin{equation}\tag{1}
\nabla(u^{+}) =
\begin{cases}
\nabla u &\text{ if }u>0 \\
0 &\text { if }u\leq0
\end{cases}
\end{equation}
and
\begin{equation}\tag{2}
\nabla(u^{-}) =
\begin{cases}
\nabla u &\text{ if }u<0 \\
0 &\text { if }u\geq0
\end{cases}
\end{equation}
almost everywhere in $\Omega$.
How to show that $u^{+},u^{-}\in W^{1,p}(\Omega)$? I know that $||u^{+}||_{p} \leq ||u||_{p}$ and $||u^{-}||_{p} \leq ||u||_{p}$ but I do not know how to show that $||\nabla u^{+}||_{p}$ and $||\nabla u^{-}||_{p}$ are bounded. Also how to show that (1) and (2) hold true? Any hint is much appreciated
Thank you very much!
It is easy to see that $u^+$ or $u^-$ have as much weak derivatives as $u$, just set
$$\Omega^+:=\{x\in\Omega: u(x)\ge 0\},\quad\Omega^-:=\{x\in\Omega: u(x)\le 0\}$$
Hence
$$\int_{\Omega} u^+(x)\partial_j\varphi(x)\, dx=\int_{\Omega} u(x)\partial_j\varphi(x)\chi_{\Omega^+}(x)\, dx=-\int_{\Omega}\partial_j u(x)\chi_{\Omega^+}(x)\varphi(x)\, dx$$
where $\partial_j u$ is the $j$-weak partial derivative of $u$, and $\varphi$ is any test function (it holds because we can choose any test function with compact support in $\Omega^+$, hence its easy to see that it holds for any test function).
Thus $\partial_j u\chi_{\Omega^+}$ is the $j$-weak partial derivative of $u^+$, hence it follows that $\nabla u^+=\nabla u\chi_{\Omega^+}$.
Thus clearly $\|\nabla u^+\|_p\le\|\nabla u\|_p$, just note that
$$|\nabla u^+(x)|^p=|\nabla u(x)|^p|\chi_{\Omega^+}(x)|^p\le |\nabla u(x)|^p$$
for all $x\in\Omega$ and all $p\ge 0$.
The same can be shown for the other case.