This is an extension to Partial differential equations by Lawrence C. Evans, page 291, Q15.
Fix $\alpha > 0$, $U$ is a bounded open connected set with smooth boundary. Show that there exists some constant $C$ such that $\forall u \in W^{1, p}(U)$ with trace $Tu = 0$ on a relatively open $\Gamma\subset\partial U$ with $\mathcal{H}^{n-1}(\Gamma)\ge \alpha$, $$\|u\|_{L^p(U)} \le C \|\nabla u\|_{L^p(U)}$$ [$\mathcal {H}^{n-1}$ is the $(n-1)$-dimensional measure on $\partial U$.]
The original question solution uses Poincare inequality, and I think this extension also uses Poincare inequality. Basically, the value of the integral of $u$ is bounded by its average value plus its oscillation, and the oscillation is bounded by its derivative through Poincare inequality. Then it remains to bound the average value of $u$ by its derivative.
Since $u$ is anchored at $0$ on $\Gamma$, the average value of $u$ can't deviate too far from zero without causing a big derivative. But how to turn this quantitative is hard.
By mimicking the solution to the original question, I have found that, letting $T: W^{1,p}(U) \rightarrow L^p(\partial U)$ be the bounded trace operator.
$$\|Tu\|_{L^p(\partial U)} \le C\|\nabla(Tu)\|_{L^p(\partial U)}$$
This does not help though, since taking the trace gives the wrong direction for the inequality
$$ \|Tu\|_{L^p(\partial U)} \le C\|u\|_{W^{1,p}(U)}$$
when I really need to bound $\|u\|_{W^{1,p}(U)}$ from above.