This is Theorem 3 in the section 5.6.1. of Evans' PDE book:
Here is also Theorem 2 in the same section:
What I don't understand is why he included that proof of Theorem 3. Isn't what he did in the proof precisely the same reasoning from the proof of Theorem 2? Furthermore, isn't the Poincare inequality also true for $q\in [1, p^{*}]$ for functions in $W^{1, p}(U)$, not just $W_0^{1, p}(U)$? I mean, in the proof of Theorem 2 we have the estimate $$||\overline{u}||_{L^{p^{*}}(U)}\le C ||D\overline{u}||_{L^p(U)}$$ and if we use the fact that $\overline{u}=u$ in $U$ and that $\overline{u}$ has compact support and then we use that $U$ is bounded we get precisely the Poincare inequality in this case. Of course, for $q>p^{*}$ we really need $u\in W_0^{1, p}(U)$, but I think that in the $q\in [1, p^{*}]$ case we may obtain this result. I was just wondering why Evans chose to include these details in the proof of Theorem 3, to me it looks like they all follow from Theorem 2.
Just to reiterate a few points from the comments:
The Poincaré inequality does not hold in $W^{1,p}(U)$, consider nontrivial constants.
The estimate $\|\bar u\|_{L^{p^*}(U)} \le C \|D \bar u\|_{L^p(U)}$ is false in general. In the proof of Theorem 2 this estimate is asserted with $U$ replaced by $\mathbb R^n$.
If $u \not\equiv 0$, $\bar u$ will never be constant in $\mathbb R^n$.
Theorem 3 does not follow from Theorem 2 but (as Theorem 2) from Theorem 1.
Theorem 3 does not need any regularity of $\partial \Omega$. (The point is that $C_c^\infty(U)$ functions do not “see” the boundary at all.)