Help me please, how can I show that Poincaré inequality doesn't hold in an unbounded domain?
Thanks a lot!
If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality holds : $\left \| u \right \|_{L^{2}(\Omega) }\leq c \left \| \nabla u \right \|_{L^{2}(\Omega) } $
where $c$ depends only on $\Omega$ and not on $u$.