On the boundary sup of Sobolev functions

Question

In Gilbarg-Trudinger's section on the maximum principle for weak solutions, the boundary sup of Sobolev functions defined as follows:

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. For $u, v \in H^1(\Omega)$, $u \leq v$ on $\partial \Omega $ if $(u-v)^+ \in H^1_0 (\Omega)$. Define $$\sup_{\partial \Omega} u = \inf \; \{ k \in \mathbb{R} \; | \; u \leq k \text{ on } \partial \Omega \}, \quad u \in H^1(\Omega).$$

Here, of course, $H^1(\Omega) = W^{1,2}(\Omega)$ and $H^1_0(\Omega) = W^{1,2}_0(\Omega)$.

My question is whether the following inequalities are valid.

Let $u, v \in H^1(\Omega)$.

1. $\sup_{\partial \Omega} (u+v) \leq \sup_{\partial \Omega} u + \sup_{\partial \Omega} v.$
2. For $k \in \mathbb R$ with $\sup_{\partial \Omega} u \leq k$, $u \leq k$ on $\partial \Omega$.
3. If $u \leq v$ on $\partial \Omega$, then $\sup_{\partial \Omega} u \leq \sup_{\partial \Omega} v.$

I tried to prove these inequalities by the definition, but it does not seem straightforward. I also looked for some questions on MathStackExchange and MathOverflow using some relevant keywords, but there are no results on these inequalities. (Addition: This might be related.)

I initially guessed that the inequalities are valid, and in fact those are true when the domain has Lipschitz boundary. The proof is done using the trace operator. See the comments and the linked Math.SE post above. However, I am not sure there is any proof without using the trace operator or without the assumption on regularity of the boundary. Could you provide any good idea?

Thanks!

P.S I posted the same question on MathOverflow (link)