$u \in H_0^1(\Omega) \implies u^+ \in H_0^1(\Omega)$ (boundary not $C^1$)

Question

I know how to prove that $u \in H^1(\Omega) \implies u^+ \in H^1(\Omega)$ since this is Exercise 5.18 in Evan's PDE book. However, I'm not sure how to extend this to $H_0^1(\Omega)$. My first thought was to use trace theorems to prove this, but apparently (?) this result also holds for functions with a boundary that is not $C^1$ . I have seen the post

Does $u \in W^{1,2}_0(\Omega)$ imply $|u| \in W^{1,2}_0(\Omega)$?

However, the solution relies on chain rule, which requires a $C^1$ boundary.

Answer 1

The following should work (some technical details are missing):

1. Approximate $u$ by a sequence $(v_n) \subset C_c^\infty(\Omega)$.
2. Mollify $v_n^+$ to obtain $w_n \in C_c^\infty(\Omega)$ or use something like $w_n = (\sqrt{v_n^2 + \varepsilon_n^2} + v_n ) / 2$.
3. Verify $w_n \to u^+$ in $H^1(\Omega)$.