In the proof of Theorem 2, chapter 2, section 5.5, from Evans's book (second edition) we have the following statements:
My doubt is how to get the relations (7) and (8). I thought of using the Theorem 2, section 5.3.2 about global aproximation, but there it is necessary that the domain be bounded. So, I don't understand how to get this result. It doesn't seem very obvious to me.
I believe the argument is as follows: The space $C^\infty(\mathbb{\overline{R^n_+}})$ is dense in $W^{1,p}(\mathbb{R}^n_+)$ because $\mathbb{R}^n_+$ has the segment property. Hence, there is a sequence $(u_m) \in C^\infty(\mathbb{\overline{R^n_+}}) \subset C^1(\mathbb{\overline{R^n_+}})$ such that
$u_m \to u$ in $W^{1,p}(\mathbb{R}^n_+)$ and by the continuity of the trace operator, together with the hypothesis that $Tu=0$, we obtain the statement (8). Right?