the constraint component of the energy strongly penalizes $\phi^2 |\s|^2$ so that $\phi$ should be small in the region where $|\sigma|$ is large. On the other hand the behavior of $\phi$ is controlled by the Modica Mortola component that forces $\phi$ to be close to $1$ away from a one-dimensional set as ${\varepsilon}$ converges to 0. As a consequence, we expect the support of $\sigma$ and the energy to concentrate on a one dimensional set connecting $S$. The main part of the paper consists in making rigorous and quantitative this analysis.

From now on, we assume that there exists some $\a\geq 0$ such that $$\label{eq:limalpha}
\dfrac\eta{{\varepsilon}}\ \stackrel{{\varepsilon}\downarrow0}\longrightarrow\ \a.$$ We note $\M_S({\overline}{\Omega})$ the set of ${\mathbf{R}}^2$-valued measures $\s\in \M({\mathbf{R}}^2,{\mathbf{R}}^2)$ with support in ${\overline}{{\Omega}}$ such that the constaint  holds. We define the limit energy ${\mathcal{E}}_\a:\M({\overline}{\Omega},{\mathbf{R}}^2)\times L^1({\Omega}){\rightarrow}[0,+{\infty}]$ as $$\label{eq:limit}
   {\mathcal{E}}_\a(\s,\phi)=
   \begin{dcases}
    \;\int_{M}(1+\a\,\t)\dH^1 & \text{if } \phi\equiv1,\ \s\in \M_S({\overline}{\Omega})\mbox{ and }\s=U(M,\t,\xi),\\
  ~  \quad+{\infty}&\text{in the other cases.}
   \end{dcases}$$ We prove the $\Gamma$-convergence of the sequence $({\mathcal{F}}_{\varepsilon})$ to the energy ${\mathcal{E}}_\a$ as ${\varepsilon}{\downarrow}0$. More precisely the convergence holds in $\M({\overline}{\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ where $\M({\overline}{\Omega},{\mathbf{R}}^2)$ is endowed with the weak star topology and $L^1({\Omega})$ is endowed with its classical strong topology.\
We establish the following lower bound.

\[teo:sigma\_liminf\] 