I am currently working through notes on Sobolev Spaces in Partial Differential Equations by Lawrence.c.Evans

Check 2 page attachment for relevant notes. It is the proof of the Trace Theorem.

In the proof I would like to know why (1) is not trivial since $\displaystyle \Gamma \subset \partial U$ and $\displaystyle \partial U$ is a boundary which therefore has measure zero, does it not follow then that $\displaystyle \int_{\Gamma} |u|^{p}dx' \leq \int_{\partial U }|u|^{p}dx' = 0$, therefore $\displaystyle \int_{\Gamma} |u|^{p}dx' = 0\text{ } \leq\text{ } C\int_{U}|u|^{p} + |Du|^{p}dx$.

And then would the last line $\displaystyle ||u||_{L^{p}(\Gamma_{i})} \text{ } \leq \text{ } C||u||_{W^{1,p}(U)} \text{ for i=1,...,N}$ not also follow easily since $\displaystyle 0 = ||u||_{L^{p}(\Gamma_{i})} \text{ } \leq \text{ } C||u||_{W^{1,p}(U)} \text{ for i=1,...,N}$.

Thanks