Poincaré inequality is given by $\int_\Omega u^2\le C\int_\Omega|\nabla u|^2dx ,$ where $\Omega$ is bounded open region in $\mathbb R^n$. However this inequality is not satisfied by all the function. Take for example a constant function $u=10$ in some region.
Happy to have have some discussions about it. Thanks for your help.