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We are trying to show that the map $f\mapsto|f|$ is a continuous (nonlinear) map from $W^{1,p}(\Omega)\to W^{1,p}(\Omega)$ for any bounded/open region $\Omega$ and for $p\in[1,\infty)$.

We have tried using the regular definition of continuity using the standard norm on Sobolev spaces for both $f$ and $|f|$:

$\displaystyle\|u\|^p=\sum_{\alpha:|\alpha|\leq1}\|\partial^\alpha u\|^p_{L^p(\Omega)}$

where $\alpha$ is a multi-index and $|\alpha|$ denotes the sum of its components, but we unable to make any progress. Is there another approach we should try, or some trick that we seem to be forgetting? Thank you for your time.

PhasedAndConfused
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    see http://math.stackexchange.com/q/1433599/254733 and http://math.stackexchange.com/q/1138797/254733 and http://math.stackexchange.com/q/384448/254733 – Svetoslav Feb 03 '16 at 20:30
  • My biggest concern (in regards to the third link), is that Stampacchia's theorem seems to only apply to only functions $v\in W^{1,p}_0 (\Omega)$ and not $v\in W^{1,p} (\Omega)$ – PhasedAndConfused Feb 03 '16 at 23:38
  • I also don't see how this helps prove continuity - these links only help to show that $|f|$ is in the correct space... – PhasedAndConfused Feb 03 '16 at 23:40
  • I think it is also applicable for $W^{1,p}$. Now I can not find the reference. – Svetoslav Feb 03 '16 at 23:40

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