Let $\Omega \subset R^n (n \geq 2)$ a bounded domain with smooth boundary. Let $\phi \in W^{1,p}(\Omega)\cap L^{\infty}(\Omega) \ (2 \leq p < \infty)$ with $\phi^{+}$ different from the null function. Let $j_0 \in N$ the smallest natural number such that $j_0 \geq \| \phi\|_{L^{\infty}(\Omega)}.$ Let $ u \in W^{1,p}(\Omega)$ with $u - \varphi \in W^{1,p}_{0}(\Omega).$
I am reading a paper and the paper says that $ (|u| - j)^{+} \in W^{1,p}_{0}(\Omega)$ for $ j \in N$ with $j > j_0$. I have no idea of how prove this. Someone could help me ?
Thanks you for your attention.
Maybe this can help someone one day. Since I saw this argument in a paper, maybe it is important.
Since $u - \varphi \in W^{1,p}_{0}(\Omega)$, we have
$|u - \varphi| \in W^{1,p}_{0}(\Omega).$
Note that $||u(x)| - |\varphi (x)|| \leq |u(x) - \varphi (x)|,$ for all $x$ then $||u| - |\varphi|| \in W^{1,p}_{0}(\Omega) $. For $j>j_0$ we have for all x that $|\varphi(x)| < j$, which implies $- |\varphi (x)| > -j$.
For all $x$ we have $\max(|u(x)| - |\varphi (x)| , 0) \geq \max(|u| - j, 0) \geq 0 \ (*)$.
Since $ |u| - |\varphi| \in W^{1,p}_{0}(\Omega) \rightarrow (|u| - |\varphi|)^{+} \in W^{1,p}_{0}(\Omega) \ (**) $. From $(*)$ and (**) we have $(|u| - j)^{+} \in W^{1,p}_{0}(\Omega) .$