Poincaré Inequality
Let $\Omega \subset \mathbb{R^n}$ be a bounded set, then $\exists c=c(\Omega) \gt 0$ s.t $||u||_{L^2(\Omega)} \le ||\nabla u||_{L^2(\Omega)}, \forall u \in H^1_0(\Omega)$
We have also a Generalization
$||u||_{L^2(\Omega)} \le ||\nabla u||_{L^2(\Omega)}, \forall u \in (H^1\bigcap L^2)_{/\mathbb{R}} \equiv H^1_{/\mathbb{R}}$
where $L^2_{/\mathbb{R}}$ $=\{u° \text{s.t } v \in u° \Rightarrow u=v+c \text{ with } c \in \mathbb{R}\}$
with
$||u°||_{L^2(\mathbb{R})}:=$ $\inf ||v||_{L^2(\Omega)}:=$ $\inf ||u+c||_{L^2}$ s.t $v \in u°, c \in \mathbb{R}$
and
$\displaystyle H^1_{/\mathbb{R}}$ $=\Big\{\displaystyle v \in H^1(\mathbb{R})$ s.t $\displaystyle\int_{\Omega} u\ =0\Big\}$
Can I have a proof for the Generalization pls?