Let $\Omega$ be a polyhedral subset of $\mathbb{R}^2$ (for example: a triangle). Prove that exists a constant $\beta>0$ such that for all $q\in L^2(\Omega)$ it holds:
$$\beta\|q\|_{0,\Omega}\leq\sup_{v\in(\mathbb{P}_{0}(\Omega)^2)^\perp}\dfrac{\int_\Omega q\,\textrm{div}(v)\, dx}{\|v\|_{1,\Omega}}$$
where $\mathbb{P}_0(\Omega)^2$ is the set of vectors in $\mathbb{R}^2$ that are constant on $\Omega$, $\|v\|_{1,\Omega}^2=\|v\|_{0,\Omega}^2+\|\nabla v\|_{0,\Omega}^2$ (the usual norm of $H^1(\Omega)$) with $\|v\|_{0,\Omega}^2=\int_\Omega v\cdot v\, dx$ (the usual norm in $L^2(\Omega)$) and the elements of $H^1(\Omega)^2$ that are orthogonals to the space $\mathbb{P}_{0}(\Omega)^2$, denoted $(\mathbb{P}_{0}(\Omega)^2)^\perp$, is with respect to the inner product of $H^1(\Omega)$:
$$(u,v)_{1,\Omega}=\int_\Omega v\cdot v\,dx+\int_\Omega \nabla v:\nabla v\,dx.$$