Suppose that $1 \leq p < \infty $ and $\Omega$ is a bounded open set. Then according to Poincare inequality, there exists a constant $C$ depending on $\Omega$ and $p$ such that
$$\|u\|_{L^p(\Omega)} \leq C\|\nabla u\|_{L^p(\Omega)}, \forall u\in W_{0}^{1,p}(\Omega)$$
Now, I want to apply Poincare inequality to functions that has non-zero boundaries, and I want the following inequality
$$\|u\|_{L^p(\Omega)} \leq C(\|\nabla u\|_{L^p(\Omega)}+\| u\|_{L^p(\partial \Omega)}), \forall u\in W^{1,p}(\Omega)$$
I have searched across the Sobolev space section in Brezis, Evans, and Maz'ya, but could not find something that exactly satisfy my need. I have found a paper titled "Best constant in Poincaré inequalities with traces", but the author displayed the equation without any proof, and I'm not sure how he get the inequality.