Is the following, I call it bilinear Poincaré inequality, true?

Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in H^2(\Omega)$, \begin{equation} \int_{\Omega} (u - \bar{u})(v - \bar{v}) \dL x \le C\int_{\Omega} \lvert Du\rvert\lvert Dv\rvert\dL x \end{equation} where $\bar{u}$, $\bar{v}$ are the mean values of $u$ and $v$ over $\Omega$ respectively.

If the above is false, is there any other form of "bilinear Poincare inequality" or "bilinear Sobolev inequality" and even for $p$ other than 2?

Is the following, I call it bilinear Poincaré inequality, true?

Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in H^2(\Omega)$, \begin{equation} \int_{\Omega} (u - \bar{u})^2(v - \bar{v})^2 \dL x \le C\int_{\Omega} \lvert Du\rvert^2\lvert Dv\rvert^2 \dL x \end{equation} where $\bar{u}$, $\bar{v}$ are the mean values of $u$ and $v$ over $\Omega$ respectively.

If the above is false, is there any other form of "bilinear Poincare inequality" or "bilinear Sobolev inequality" and even for $p$ other than 2?

Is the following, I call it bilinear Poincaré inequality, true?

Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in H^2(\Omega)$, \begin{equation} \int_{\Omega} (u - \bar{u})(v - \bar{v}) \dL x \le C\int_{\Omega} \lvert Du\rvert\lvert Dv\rvert\dL x \end{equation} where $\bar{u}$, $\bar{v}$ are the mean values of $u$ and $v$ over $\Omega$ respectively.

Is the following, I call it bilinear Poincaré inequality, true?

Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in H^2(\Omega)$, \begin{equation} \int_{\Omega} (u - \bar{u})(v - \bar{v}) \dL x \le C\int_{\Omega} \lvert Du\rvert\lvert Dv\rvert\dL x \end{equation} where $\bar{u}$, $\bar{v}$ are the mean values of $u$ and $v$ over $\Omega$ respectively.

If the above is false, is there any other form of "bilinear Poincare inequality" or "bilinear Sobolev inequality" and even for $p$ other than 2?

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

Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in H^2(\Omega)$, \begin{equation} \int_{\Omega} (u - \bar{u})(v - \bar{v}) \dL x \le C\int_{\Omega} |Du||Dv|\dL x \end{equation} where $\bar{u},\bar{v}$ are the mean values of $u$ and $v$ over $\Omega$ respectively.

Is the following, I call it bilinear Poincaré inequality, true?

Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in H^2(\Omega)$, \begin{equation} \int_{\Omega} (u - \bar{u})(v - \bar{v}) \dL x \le C\int_{\Omega} \lvert Du\rvert\lvert Dv\rvert\dL x \end{equation} where $\bar{u}$, $\bar{v}$ are the mean values of $u$ and $v$ over $\Omega$ respectively.