Let $\Omega\subset\mathbb{R}^N$ be a bounded and connected Lipschitz domain, and $u\in H^1(\Omega)$ be any function. If I denote for each real $k\in\mathbb{R}$
$$\Omega_k=\{x\in\Omega\ |\ u(x)>k\}$$
how can I rigurously prove that $(u-k)^+:=\max\{u-k,0\}\in H^1_0(\Omega_k)$ for any $k$ with $|\Omega\setminus\Omega_k|>0$ (Lebesgue measure)?
I know that $(u-k)^+\in H^1(\Omega)$ and $(u-k)^+\in H^1(\Omega_k)$ as a consequence, but I don't know how to prove the above fact...