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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?

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  • $\begingroup$ Check out this paper: arxiv.org/pdf/math/0601667.pdf $\endgroup$ – Chee Han Mar 20 '17 at 17:26
  • $\begingroup$ Hey. Thanks, I checked it out but I can't find anyhting that helps? What should I be looking at? Cheers. $\endgroup$ – Henrymerrild Mar 20 '17 at 19:09
  • $\begingroup$ It is example 3.6 in the linked paper $\endgroup$ – daw Mar 20 '17 at 19:17
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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 $$ \|u_{\partial\Omega}\|_{L^p(\Omega)}\le K\|u\|_{W^{1,p}(\Omega)}\quad \forall u\in W^{1,p}(\Omega). $$

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