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.