mean value and let $(\phi_{\varepsilon})\subset H^1(O)$ such that $\phi_{\varepsilon}\in H^1(O,[\eta({\varepsilon}),1])$. Assume that $c_0:=\sup_{\varepsilon}\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)$ is finite, then there exist a subsequence ${\varepsilon}_j$ and a function $u\in BV({\Omega})$ such that

a)  $\phi_{{\varepsilon}_j} {\rightarrow}1$ in $L^2(O)$,

b)  $u_{{\varepsilon}_j}{\rightarrow}u$ with respect to the weak-$*$ convergence in $BV$,

c)  $u\in \P(O)$.

Furthermore for any $u\in\P(O)$ and any sequence $(u_{\varepsilon},\phi_{\varepsilon})$ as above such that $u_{\varepsilon}\rwstar u$, we have the following lower bound of the energy: $$\liminf_{{\varepsilon}{\rightarrow}0}\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)\geq \int_{J_u\cap O}{\left}[1+\alpha|[u]|{\right}]\dH^{d-1}.$$

The proof is achieved in several steps and mostly follows ideas from [@Iur] (see also [@ContiFocardiIurlano]). In the first step we obtain *(a)* and *(b)*. In steps 2. we prove *(c)* and the lower bound for one dimensional slices. Finally in step 3. we prove *(c)* and the lower bound in dimension $d$. The construction of a recovery sequence that would complete the $\Gamma$-limit analysis is postponed to the global model in Section \[sec:UppBound\].

*Step 1.* Item *(a)* is a straightforward consequence of the definition of the functional. Indeed, we have $$\int_O(1-\phi_{\varepsilon})^2\, \dx\, \leq {\varepsilon}\, \LF_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\, \leq\, c_0\, {\varepsilon}\,  \stackrel{{\varepsilon}{\downarrow}0}\longrightarrow \ 0.$$ For *(b)*, since $(u_{\varepsilon})$ has zero mean value, we only need to show that $ \sup_{\varepsilon}\{|Du_{\varepsilon}|(O): k\in{\mathbf{N}}\}<+\infty$. Using Cauchy-Schwarz inequality we get $$\label{eq:compactness1}
{\left}[|Du_{\varepsilon}|(O){\right}]^2={\left}(\int_{O}|{\nabla}u_{\varepsilon}|{\right})^2\leq {\left}({\varepsilon}\int_{O}\frac{1}{\phi^2\e}{\right}){\left}(\frac{1}{{\varepsilon}}\int_{O}\phi^2\e|{\nabla}u_{\varepsilon}|^2{\right}).$$ By 