Given $u \in W^{1,p}(\Omega)$ and $\alpha$ a function which verifies $\alpha \in C^1(\Omega)$, $\alpha \in L^{\infty}(\mathbb{R}^n)$, $\nabla \alpha \in L^{\infty}(\mathbb{R}^n)^n $ y $supp(\alpha) \subset \mathbb{R}^n-{\partial \Omega}$.
I need to show that:
$\overline{\alpha u} \in W^{1,p}(\mathbb{R}^n)$ and $\overline{(\alpha u)}_{x_i}= \overline{\alpha u_{x_i}+\alpha_{x_i}u}$
Where $\bar{f}$ is the extension of a given function $f$, defined as $$\bar{f}(x)=\left\{\begin{matrix} f(x) & in \ \Omega \\ 0 & \quad \quad \ in \ \mathbb{R}^n-\Omega \end{matrix}\right. $$ And $f_{x_i}$ denotes $\frac{\partial f}{\partial x_i}$.
I have tried several things, the last one being the one I present here, although it is not completed:
Recall the definition of weak derivative, and pick $\varphi \in C^1_c(\mathbb{R}^n)$. Then:
$\int_{\mathbb{R}^n}\overline{\alpha u} \varphi_{x_i}= \int_{\Omega}\alpha u \varphi_{x_i}=\int_{\Omega}u[(\alpha \varphi)_{x_i}- \alpha_{x_i} \varphi]$
Now it would be great if I could show that $\alpha \varphi$ is actually in $C^1_c(\Omega)$, which would allow us to pass the derivative to $u$ in the first term, but I do not know exactly how to proceed.