# Can we obtain the following variation of Poincaré inequality?

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

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