Let $\Omega$ be a bounded domain over $\mathbb{R}^N$ and $u:\Omega\to\mathbb{R}$ is a function.
Then for any $1<p<\infty$, we have
$$
\int_{\Omega}|\nabla|u||^p\,dx\leq\int_{\Omega}|\nabla u|^p\,dx,
$$
provided both the integrals exist.
Indeed, we have
$$
\int_{\Omega}|\nabla|u||^p\,dx=\int_{\Omega\cap\{u>0\}}|\nabla u|^p\,dx+\int_{\Omega\cap\{u\leq 0\}}|\nabla (-u)|^p\,dx=\int_{\Omega\cap\{u>0\}}|\nabla u|^p\,dx+\int_{\Omega\cap\{u\leq 0\}}|-\nabla u|^p\,dx=\int_{\Omega\cap\{u>0\}}|\nabla u|^p\,dx+\int_{\Omega\cap\{u\leq 0\}}|\nabla u|^p\,dx=\int_{\Omega}|\nabla u|^p\,dx.
$$
So, I get equality.
Can somebody please confirm, if this argument seems fine. Thanks.
Best Answer
If all is well-defined, I do not see any big issue. Note that, a priori, we may have that $ u$ is differentiable, while $|u|$ not. But it this is not the case, the argument is fine.