Let $u\in H^1(\mathbb{R}^n)$ where $H^1(\mathbb{R}^n)$ is the standard Sobolev space $W^{1,2}(\mathbb{R}^n)$. We already know that if $\Omega$ is a bounded domain, then for any $u\in H^1(\Omega)$, there must holds $u^+,\,u^-,\,|u|\in H^1(\Omega)$. Then it is trivial to know that $$ \int_{\Omega}|\nabla u|^2=\int_{\Omega}|\nabla |u||^2. $$ My question is that does this property still holds for unbounded domain? Especially, if we consider the case $\mathbb{R}^n$, does $|u|\in H^1(\mathbb{R}^n)$ and $$ \int_{\mathbb{R}^n}|\nabla u|^2=\int_{\mathbb{R}^n}|\nabla |u||^2\,\,? $$ Here is my attempt: Since $u\in H^1(\mathbb{R}^n)$, then $u\in H^1(\Omega)$ for any bounded domain $\Omega$. Then we know $|u|\in H^1(\Omega)$. Hence for any test function $\phi\in C_0^\infty(\mathbb{R}^n)$, there must exists some $\Omega\subset\mathbb{R}^n$ such that $supp(\phi)\subset\Omega$, then the integral by parts formula holds, and I think we can just conclude that $|u|\in H^1(\mathbb{R}^n)$. The $L^2$ norm equality for the gradient $\nabla u$ and $\nabla|u|$ is simple I think, because there hold $u=u^+-u^-$, $|u|=u^++u^-$ and $u^+\cdot u^-\equiv 0$.
Am I right? I think this is a standard attempt but I just can't find a precise theorem for the case of $\mathbb{R}^n$. If there are any errors in the above proof, please just point them out. Thanks for any help!