Let be $u\in W^{1,2}(U)$, where $U\subset \Bbb R^n$ is an bounded open. Define $$\sup_{\partial U} u_+:=\inf \{\ l\in \Bbb R: (u_+-l)_+\in W_0^{1,2}(U)\}.$$why the set is not empty?If $k\in \Bbb R$ is such that $$\sup_{\partial U} u_+\le k<\sup_U u$$ then why $(u-k)_+\in W^{1,2}_0(U)?$
The question is easy if $u\in C^1(\overline{U})$ but why it is right in general?