In Evans' PDE, there is Trace Theorem in Chapter 5. Everything is fine except for when $p=1$ since I doubt if I could use Gauss-Green Theorem and differentiation. More specifically, a part of the proof goes on as follows
$$\int_{\Gamma} \vert u \vert^p \,dx' \le \int_{\{x_n = 0\}} \zeta\vert u \vert^p \,dx' = -\int_{B^+}(\zeta \vert u \vert^p)_{x_n}\,dx.$$
(where $B$ is an open ball with its center on $\{x_n =0\}$, $B^+$ is the upper part of $B$ and $\zeta$ is a cut-off function with $0$ outside of $B^+$)
The second equality is the Gauss-Green theorem. If $ p >1$, then everything works fine. However, if $p=1$, can the theorem and the $(\zeta \vert u \vert^p)_{x_n}$ be guaranteed? Can we stick to this method with a little modification? I'd appreciate any help!