|\s_{\varepsilon}|({\Omega})\leq\sqrt{16\frac{{\varepsilon}^2}{\eta^2}\:c_0^2 + 8{\varepsilon}|{\Omega}| c_0} \,\stackrel{{\varepsilon}{\downarrow}0}\longrightarrow\,  \frac{4 c_0}{\a}\,<\,\infty.$$ Thus the total variation of $(\s_{\varepsilon})$ is uniformly bounded and there exists $\s\in \M_S({\overline}{{\Omega}})$ such that up to extraction $\s_{\varepsilon}\to\s$ weakly-$*$ in $\M({\overline}{\Omega})$.

Now, considering the last term in the energy  we have $$\int_{\Omega}(1-\phi_{\varepsilon})^2 \dx\leq 2{\varepsilon}\;{\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\leq 2{\varepsilon}\;c_0 \rightarrow 0.$$ Hence, $\phi_{\varepsilon}\to1$ in $L^2({\Omega})$.

Let us now study the structure of the limit measure $\s$. Let us recall that $\hat{\Omega}$ is a bounded convex open set such that ${\overline}{\Omega}\subset\hat{\Omega}$ and let us extend $\s_{\varepsilon}$ by 0 and $\phi_{\varepsilon}$ by 1 in $\hat{\Omega}\setminus{\overline}{\Omega}$. Obviously we have ${\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon};\hat{\Omega})={\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon};{\Omega})$, therefore for any $O\in\A_s(\hat{\Omega})$ applying the localization described in Section \[sec:LocRes\] we can associate to each $\s_{\varepsilon}$ a function $ u_{\varepsilon}\in H^1(O)$ with mean value $0$ such that $\s_{\varepsilon}={\nabla}^\perp u_{\varepsilon}$ in $O$. By Theorem \[teo:localResult\] there exists $u\in\P(O)$ such that up to extraction $u_{\varepsilon}\rwstar u$. Eventually, by uniqueness of the limit, we get $$\s{ \!\!\mbox{{ \Large$\llcorner$}} }O=-[u]\nu_{J_u}^\perp\H^1{ \!\!\mbox{{ \Large$\llcorner$}} }(J_u\cap O).$$ Since we can cover ${\overline}{\Omega}\setminus S$ by finitely many sets $O\in \A_s(\hat{\Omega})$, this shows that $\s$ decomposes as $$\s\, =\, U(M_\s,\t_\s,\xi_\s)+\underbrace{\sum_{j=0}^N c_j \delta_{x_j}}_{\mu}.$$ By Lemma \[lemma:completediv\] there exists a rectifiable measure $\g=U(M_\g,\t_\g,\xi_\g)$ such that ${\nabla}\cdot(\s+\g)=0$ and $\H^1(M_\g\cap M_\s )=0$. Then there exists $u\in BV({\Omega})$ such that $Du 