$e^{i\varphi}=u\in \mathcal{M}_{div}(\Omega)$. Furthermore the authors are conjecturing that the $\Gamma$-Limit should be given by the following functional $E_0$ over ${\mathcal M}_{div}(\Omega)$ : $$E_0(u):=2\int_{a\in\mathbb{R}}\left|\operatorname{div}\left(e^{i\varphi\wedge a}\right)\right|(\Omega)\,da$$ Part of the $\Gamma-$convergence has been proved as they established in one hand the following inequality $$E_0(u):=2\int_{a\in\mathbb{R}}\left|\operatorname{div}\left(e^{i\varphi\wedge a}\right)\right|(\Omega)\,da\leq\liminf E_{\varepsilon_{n'}}(u_{\varepsilon_{n'}})$$ and in the other hand that $$\label{inf_energy}
\lim_{\varepsilon\to0}\inf_{u\in W^{1,2}} E_{\varepsilon}(u)=2\inf_{u\in\mathcal{M}_{div}(\Omega)}\int_{a\in\mathbb{R}}\left|\operatorname{div}\left(e^{i\varphi\wedge a}\right)\right|(\Omega)\,da=2|\partial\Omega|\,,$$ where $|\partial\Omega|$ is the perimeter of the set $\Omega$. One can prove (see [@RS1]), that the infimum on the right hand side is achieved by $u=-\nabla^{\perp}\operatorname{dist}(\cdot,\partial\Omega)\in\mathcal{M}_{div}(\Omega)$. The function $g=\nabla^{\perp}\operatorname{dist}(\cdot,\partial\Omega)$ is the viscosity solution of $$\label{eikonal}
\left.\begin{array}{rclcl}
|\nabla g|-1&=&0&\mbox{on}&\Omega\,,\\
g&=&0&\mbox{on}&\partial\Omega\,.
\end{array}\right\}$$ A question, which was left open in [@RS1] and [@RS2] was to describe the possible limits $u$ of minimizing sequence of (\[energy\]). It was conjectured that $u=\pm\nabla^{\perp}dist(\cdot,\partial\Omega)$ are the only possible limits of sequences of minimizers. A positive answer to this conjecture has been given in [@ALR]. Precisely, in [@RS2] it is proved that the limit $u$ of a minimizing sequence of (\[energy\]) satisfies the [*entropy condition*]{} $$\label{positive_limit}
\operatorname{div}e^{i\varphi\wedge a}\geq0\quad\mbox{for all}\quad a\in\mathbb{R}$$ or $\operatorname{div}e^{i\varphi\wedge a}\leq0\quad\mbox{for all}\quad a\in\mathbb{R}$. Then in [@ALR] the following result is established

[@ALR] Let $u=-\nabla^{\perp}g$ be a divergence free unit vector-field in the space $\mathcal{M}_{div}(\Omega)$. The entropy condition (\[positive\_limit\]) holds if and only 