assumption, the second therm in the right hand side of is bounded by $2c_0$. In order to estimate the first term we split $O$ in the two sets $\{\phi_{\varepsilon}<1/2\}$ and $\{\phi_{\varepsilon}\geq1/2\}$. We have,$${\varepsilon}\int_{O}\frac{1}{\phi\e}\,=\, {\varepsilon}\int_{\{\phi_{\varepsilon}<1/2\}}\frac{1}{\phi^2\e}+{\varepsilon}\int_{\{\phi\e\geq1/2\}}\frac{1}{\phi^2\e}.$$ Since $\phi_{\varepsilon}\geq \eta$ and $(1-t)^2$ is decreasing in the interval $O$ the following inequalities hold $$\begin{aligned}
\int_{\{\phi\e<1/2\}}\frac{1}{\phi^2\e}&\leq& \frac{2 {\varepsilon}}{\eta^2(1-1/2)^2}\int_{O}\frac{(1-\phi\e)^2}{2{\varepsilon}}\,\leq\,\frac {8{\varepsilon}}\eta^2 c_0,\\
\nonumber\int_{\{\phi_{\varepsilon}\geq1/2\}}\frac{1}{\phi^2\e}&\leq &\int_{O}\frac{1}{(1/2)^2} \, =\, 4|O|.\end{aligned}$$ Combining these estimates with  we obtain $$\label{eq:limitatezzagradu}
{\left}[|Du_{\varepsilon}|(O){\right}]^2\, \leq\, \frac{{\varepsilon}^2}{\eta^2}\:16c_0^2 + 8{\varepsilon}|O| c_0\,\stackrel{{\varepsilon}{\downarrow}0}\longrightarrow\,  \frac{16c_0^2}{\a^2}\,<\,\infty.$$ This establishes *(b)*.

*Step 2.* In this step we suppose $O$ to be an interval of ${\mathbf{R}}$. We first prove that $u$ is piecewise constant. The idea is that in view of the constraint component of the energy, variations of $u_{\varepsilon}$ are balanced by low values of $\phi_{\varepsilon}$. On the other hand the Modica-Mortola component of the energy implies that $\phi_{\varepsilon}\simeq 1$ in most of the domain and that transitions from $\phi_{\varepsilon}\simeq 1$ to $\phi_{\varepsilon}\simeq 0$ have a constant positive cost (and therefore can occur only finitely many times).

*Step 2.1. proof of $u\in \P(O)$.* Let us define $$\label{def:insiemi}
B_{\varepsilon}:= {\left}\{x\in O:\phi_{\varepsilon}(x)<\frac{3}{4}{\right}\}\,\supset\,  A_{\varepsilon}:= {\left}\{x\in O:\phi_{\varepsilon}(x)<\frac{1}{2}{\right}\},$$ and let $$\label{def:C}
C_{\varepsilon}= \{I \text{ connected component of } B_{\varepsilon}: I\cap A_{\varepsilon}\neq \varnothing \}.$$ Let us show that the cardinality 