I want to show the following statement:
if $u\in W^{1,p}(\Omega)$, then $u_-$,$u_+$ and $|u|\in W^{1,p}(\Omega)$ with $D(u_+) = Du\cdot I_{u>0}\qquad\text{and}\qquad D(|u|)=Du\cdot \text{sign}(u)\qquad \text{a.e.}$ Furthermore, if $u,\, v \in W^{1,p}(\Omega)$, then $\max\{u,v\} \in W^{1,p}(\Omega)$.
My hunch: the main difficulty is to show $u_+ \in W^{1,p}(\Omega).$ The rest can be shown by vector space property: $u_- = -u+u_+,\\ |u| = u_-+u_+,\\ \max\{u,v\} = \frac{u+v}{2}+\frac{|u-v|}{2}.$
To show $u_+ \in W^{1,p}(\Omega)$, I construct an estimation function: $u_\epsilon = (\sqrt{u^2+\epsilon^2}-\epsilon)\cdot I_{u>0}.$
My first question:
How can I show such an estimation function is in $W^{1,p}(\Omega)$ for any $\epsilon>0$?
My second question:
How to use the estimation function to show $u_+ \in W^{1,p}(\Omega)$? By sending $\epsilon \downarrow 0$?
My third question:
Could it be true for general $W^{1,p}(\Omega)$?