be achieved by this relaxation procedure.

The result of A.Poliakovsky has been inspired by previous works establishing a link between some variational optimality condition of a relaxed problem and the entropy condition at the limit. Among these works we can quote [@RS1], [@RS2] and [@ALR]. Let us describe the results established in this 3 works here :

We consider for a bounded domain $\Omega\subset\mathbb{R}^2$ the space $\mathcal{M}_{div}(\Omega)$, which consists of unit vectorfields $u$ such that $u=e^{i\varphi}$ for a $\phi\in L^{\infty}(\Omega,\mathbb{R})$ and $\operatorname{div}e^{i\varphi\wedge a}$ is a Radon measure over $\Omega\times {\R}$. This space $\mathcal{M}_{div}$ was introduced by S. Serfaty and the second author in [@RS1] and [@RS2] in connection to a problem related to micromagnetism. We give here a brief description. Let $\Omega$ be a bounded and simply connected domain, for $u\in W^{1,2}(\Omega,\mathbb{S}^1)$ and a $\varepsilon>0$ we consider $$\label{energy}
E_{\varepsilon}(u)=\varepsilon\int_{\Omega}|\nabla u|^2 + \frac{1}{\varepsilon}\int_{\mathbb{R}^2}|H|^2\,,$$ where $H=\nabla(G*\hat{u})\,, \hat{u}=u$ on $\Omega$ and $\hat{u}=0$ in $\Omega^c$ and $G$ is the kernel of the Laplacian on $\mathbb{R}^2$.

It was proved in [@RS1], [@RS2] that from any sequence $u_{\varepsilon_n}\in W^{1,2}(\Omega,\mathbb{S}^1)$ such that $\varepsilon\to0$ and $E_{\varepsilon_n}(u_{\varepsilon_n})<C$ one can extract a subsequence $u_{\varepsilon_{n'}}$ such that $\varphi_{\varepsilon_{n'}}$ converges strongly in $L^p(\Omega)$ for any $p<\infty$ to a limit $\varphi$ such that 