The Poincare theorem states that: If $\Omega$ is a bounded, connected subset of $\mathbb{R}^n$ with a $C^1$-boundary, then there exists a constant $C > 0$ (depending only on $\Omega$ and $p$) such that
$||u-u_{\Omega}||_{L^p(\Omega)} \leq C ||\nabla u||_{L^p(\Omega)}$, $\forall u \in W^{1,p} (\Omega)$,
where
$u_\Omega = |\Omega|^{-1} \int_{\Omega} u(y) dy$.
My question is the following: Is it possible to extend this to theorem naturally to include the boundary? That is, if $\Omega$ satisfies the above conditions and $u \in W^{1,p}(\Omega)$ ($1 \leq p < \infty$), do we know that there exists a constant $C > 0$ such that:
$||u - u_{\partial \Omega} ||_{L^p (\Omega)} \leq C||\nabla u||_{L^p (\Omega)}$,
where $u_{\partial \Omega} = |\partial \Omega|^{-1} \int_{\partial \Omega} [Tu] (y) dy$ and $T: W^{1,p} (\Omega) \rightarrow L^p (\partial \Omega)$ is the trace operator?
Here is the argument of the paper: Write $$ u- u_{\partial \Omega} = u-u_\Omega + u_\Omega - u_{\partial \Omega} =u-u_\Omega + (u_\Omega - u)_{\partial \Omega} $$ where $(v)_{\partial\Omega}$ denote the mean of $v$ on $\partial \Omega$. Hence $$ \|u- u_{\partial \Omega}\|_{W^{1,p}(\Omega)} \le C (1 + K) \|\nabla u\|_{L^p(\Omega)}, $$ where $C$ and $K$ are the constants in the following two inequalities: $$ \|u-u_\Omega \|_{W^{1,p}(\Omega)} \le C \|\nabla u\|_{L^p(\Omega)}\quad \forall u\in W^{1,p}(\Omega) $$ and $$ \|u_{\partial\Omega}\|_{L^p(\Omega)}\le K\|u\|_{W^{1,p}(\Omega)}\quad \forall u\in W^{1,p}(\Omega). $$