of $C_{\varepsilon}$ is bounded by a constant independent of ${\varepsilon}$. Let ${\varepsilon}$ be fixed and consider an interval $I\in C_{\varepsilon}$. Let $a,b\in \bar I$ such that $\{\phi_{\varepsilon}(a),\phi_{\varepsilon}(b)\}=\{1/2,3/4\}$. Using the usual Modica-Mortola trick, we have $$\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};I)\,\geq\,\int_I {\varepsilon}|\phi'_{\varepsilon}|^2  + \frac{(1-\phi_{\varepsilon})^2}{4{\varepsilon}} \dx \geq \int_{(a,b)} |\phi'_{\varepsilon}|(1-\phi_{\varepsilon})\dx\,\geq\, \int_{1/2}^{3/4} (1-t)\,dt \, =\, \dfrac3{2^5}.$$ Since all the elements of $C_{\varepsilon}$ are disjoint, we deduce from the energy bound that $$\# C_{\varepsilon}\leq2^5c_0/3,$$ where we note $\# C_{\varepsilon}$ the cardinality of $C_{\varepsilon}$. Next, up to extraction we can assume that $\# C_{\varepsilon}=N$ is fixed. The elements of $ C_{\varepsilon}$ are written on the form $I^{\varepsilon}_i=(m^{\varepsilon}_i-w_i^{\varepsilon},m^{\varepsilon}_i+w_i^{\varepsilon})$ for $i=1,\cdots,N$ with $m^{\varepsilon}_i<m^{\varepsilon}_{i+1}$. Since $\phi_{\varepsilon}{\rightarrow}1$ in $L^1(O)$ we have $$\label{misura:zero}
\sum_{I^{\varepsilon}_i\in C_{\varepsilon}}|I^{\varepsilon}_i|= \sum_i2w^{\varepsilon}_i {\rightarrow}0.$$ Up to further extraction, we can assume that each sequence $(m_i^{\varepsilon})$ converges in ${\overline}O$. We call $m_1\leq m_2 \leq \dots \leq m_N$ their limits. We now prove that $|D u|(O\setminus \{m_i\}_{i=1}^N) =0$. For this, we fix $x\in O\setminus\{m_i\}_{i=0}^N$ and establish the existence of a neighborhood $B_\delta(x)$ of $x$ for which $|Du|(B_\delta(x))=0$. Let $0<\delta\leq 1/2\min_i|x-{m}_i|$. Equation  ensures that for ${\varepsilon}$ small enough $B_\delta(x)\cap C_{\varepsilon}=\varnothing$. Notice that from the definitions in  and  we have that $\phi_{\varepsilon}\geq 1/2$ outside $C_{\varepsilon}$. Hence, using Cauchy-Scwarz inequality, we have for ${\varepsilon}$ small enough, $${\left}(\int_{B_\delta(x)}| u'_{\varepsilon}|\dx{\right})^2\leq 