Is there a version of the Poincaré inequality which gives the estimate $$\int_\Omega |u-(u)_\Omega|^2~dx\le C\left(\int_\Omega |Du|~dx\right)^2\quad (u\in H^1(\Omega))$$ for a bounded, open domain $\Omega\subset \mathbb{R}^n$ with Lipschitz boundary, where $(u)_\Omega= \frac{1}{\mathcal{L}^n(\Omega)}\int_\Omega u ~dx$ denotes the average of $u$? The version I'm familiar with is the standard estimate $$\|u-(u)_\Omega\|_{L^p}\le C\|Du\|_{L^p}\quad (u\in W^{1,p}(\Omega)),$$ and I'm unsure how to obtain the RHS above.
Specifically, this question comes from p. 353 of this paper by Ambrosio, De Lellis, and Mantegazza. Here they consider the functionals $$F_\epsilon(u)= \frac{1}{2} \int_\Omega \left[\frac{1}{\epsilon}(1-|\nabla u|^2)^2 + \epsilon|\nabla^2 u|^2\right]~dx\quad (\Omega\subset \mathbb{R}^n).$$ Setting $v=|\nabla u|,$ they show that $$F_\epsilon(u)\ge \int_\Omega |1-v^2||\nabla v|~dx= \int_\Omega |\nabla(\Phi\circ v)|~dx,$$ where $\Phi(t)=\int_0^t |1-\tau^2|~d\tau.$ Then, by the Poincaré inequality, they give the estimate $$\int_\Omega |\Phi\circ v-m|^2 ~dx\le CF_\epsilon^2(u),$$ where $m= \frac{1}{\mathcal{L}^n(\Omega)}\int_\Omega \Phi\circ v~dx$ is the mean value of $\Phi\circ v$ on $\Omega,$ but it is unclear to me how the RHS is obtained.