Can we obtain the following variation of Poincaré inequality?

Question

The known Poincaré inequality says that in the conditions of the theorem we have \begin{equation} \|u - u_{\Omega}\|_{L^{p}(\Omega)} \le C \| \nabla u \|_{L^{p}(\Omega)}. \end{equation} see for instance [1] Can we obatain also \begin{equation} \|u - u_{\Omega}\|_{L^{p}(\Omega)} \le C \| \nabla u - (\nabla u)_{\Omega} \|_{L^{p}(\Omega)}. \end{equation}

[1] http://en.wikipedia.org/wiki/Poincar%C3%A9_inequality

Answer 1

No; take $u(x) = a\cdot x$ for some constant vector $a$. Then the right side vanishes but the left does not.