A proof of this theorem can be found in Evans (p.272, 2nd Ed.). This question addresses persons who are familiar with the proof, since I have not been able to formulate a question that generalizes my problem.
THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator $$T : W^{1,p}(U) \rightarrow L^p(\partial U)$$ such that
$\quad$(i) $Tu=u|_{\partial U}$ if $u \in W^{1,p}(U) \cap C(\bar{U})$
and
$\quad$(ii) $$\|Tu\|_{L^p(\partial U)} \le C \| u\|_{W^{1,p} (U)},$$ for each $u \in W^{1,p}(U)$, with the constant $C$ depending only on $p$ and $U$.
The problem lies in the second part (2.) of the proof.
Applying estimate (1) and changing variables, we obtain the bound $$ \int_\Gamma |u|^p dS \leq C \int_U |u|^p + |Du|^p dx. $$
Let $\Gamma$ denote the portion of the boundary of $U$ and let $\Gamma' = \Phi(\Gamma)$ the straightened portion of the boundary.
Using the change of variables formula we calculate $$ \int_\Gamma |u|^p dS = \int_{\Gamma'} |u \circ \Phi|^p dS' \leq \int_{\Phi(U)} |u \circ \Phi|^p + |D(u \circ \Phi)|^p dx $$
The first part can be transformed back as usual but we don't know how to transform the second part involving $D(u \circ \Phi)$.
