Is the following, I call it bilinear Poincare inequality,  true:

There exists $C > 0$ such that for any $u, v \in H^2(\Omega)$,
\begin{equation}
\int_{\Omega} (u - \bar{u})(v - \bar{v}) dx \le C\int_{\Omega} |Du||Dv|dx
\end{equation}
where $\bar{u},\bar{v}$ are the mean values of $u$ and $v$ over $\Omega$ respectively.