Let $\Omega$ be an open 1-regular subset of $\mathbb{R}^n$. Given $ u,v \in H^1(\Omega)\cap L^\infty(\Omega)$, how can you prove that $uv \in H^1(\Omega) $ and $\partial_j(uv)=v\partial_j u +u\partial_j v$ almost everywhere for every $j=1,\dots,n$ ?
I already proved that $uv,u\partial_j v ,v\partial_j u\in L^2(\Omega) $ (and therefore $v\partial_j u +u\partial_j v \in L^2(\Omega)$) but I'm having trouble proving that
\begin{equation} \int_\Omega (v\partial_j u +u\partial_j v) \varphi=-\int_\Omega uv \partial_j \varphi \ \ \ \ \forall \varphi \in C_0^\infty(\Omega) \end{equation}
We know that $C_0^\infty(\overline\Omega)$ is dense in $H^1(\Omega)$ so I took $\lbrace u_n \rbrace_{n\geq 1},\lbrace v_n\rbrace_{n\geq 1}\subset C_0^\infty(\overline\Omega)$ such that $\left|\left|u_n-u \right| \right|_{1,\Omega}\rightarrow 0$ and $\left|\left|v_n-v \right| \right|_{1,\Omega}\rightarrow 0$ but I'm stuck in proving that $\left| \left| v_n\partial_j u -v\partial_j u\right|\right|_{0,\Omega}\rightarrow 0$ and $\left| \left| u_n\partial_j u -u\partial_j v\right|\right|_{0,\Omega}\rightarrow 0$. If I'm I able to prove that then the result will come out easily.