Consider the following Sobolev spaces
${\bf H}^1(\Omega)= \{{\bf u } \in {\bf L}^2(\Omega) : \nabla {\bf u}\in {\bf L}^2(\Omega) \}$, with inner product $(\bf u,\bf v)_{{\bf H}^1} = \int_{\Omega} (\bf u\bf v + \nabla \bf u \nabla \bf v) dx$ and norm $\|\bf u\|_{{\bf H}^1} = (\|\bf u\|^2 + \|\nabla \bf u \|^2)^{1/2}$.
The spaces
${\bf H}^1_0(\Omega)= \{{\bf u}\in {\bf H}^1(\Omega)\,:\,{\bf u}=0\,\mbox{ on }\Gamma\}$ where $\Gamma$ is boundary of $\Omega$. With inner product $({\bf u},{\bf v})_{{\bf H}^1_0} = ( \nabla {\bf u}, \nabla {\bf v})$ and norm $\|{\bf u}\|_{{\bf H}^1_0} = \| \nabla {\bf u}\|$.
${\bf V}= \{{\bf u}\in{\bf H}^1_0(\Omega)\,:\,{\rm div}{\bf u}=0\mbox{ en }\Omega\} \ \mbox{ with norm } \|{\bf u}\|_{\bf V}=\|\nabla{\bf u}\|$
Further, the Poincaré inequality
Let $p$, so that $1 ≤ p < \infty$ and $\Omega$ a subset with at least one bound. Then there exists a constant $C$, depending only on $\Omega$ and $p$, so that, for every function $u$ of the $W^{1,p}(\Omega)$ Sobolev space,
$${\displaystyle \|u\|_{L^{p}(\Omega )}\leq C\|\nabla u\|_{L^{p}(\Omega )}}$$
Now, how is the demonstration of the equivalence of norms $\|{\bf u}\|_{{\bf H}^1_0} \equiv \| \nabla {\bf u}\|$ and $\|{\bf u}\|_{\bf V} \equiv \|\nabla{\bf u}\|$ using Poincaré inequality?
