Let $u\in H^1(\Omega )$. Show that $u^+,u^-,|u|\in H^1(\Omega )$ where $\Omega \subset \mathbb R^d$ be a smooth domain.
First of all, I know that $$W^{1,p}(\Omega )=\left\{f\in L^p(\Omega )\mid \exists g_i\in L^p(\Omega ), i=1,...,d : \forall \varphi\in \mathcal C^1_0(\Omega ),\int_\Omega f\frac{\partial \varphi}{\partial x_i}=-\int_\Omega g_i\varphi\right\}.$$
Quest 1) Can $W^{1,p}(\Omega )$ be seen as $$\left\{f\in L^p(\Omega )\mid \exists g\in L(\Omega )^d: \forall \varphi\in\mathcal C^1_0(\Omega ),\int_\Omega f\nabla \varphi=-\int_\Omega g\varphi\right\}$$ or it looks weird ?
My attempts to my problem
Let $u\in H^1(\Omega ):= W^{1,2}(\Omega )$. Then, there is $g_1,...,g_d\in L^2(\Omega )$ s.t. $$\int_{\Omega }g_i \varphi=\int_\Omega u\frac{\partial \varphi}{\partial x_i}=\int_\Omega u^+\frac{\partial \varphi}{\partial x_i}-\int_\Omega u^-\frac{\partial \varphi}{\partial x_i}$$ for all $\varphi\in \mathcal C^1_0(\Omega )$ and all $i=1,...,d$.
Q2) How can I continue ?