<\lambda<1$ the inequalities: $$\int_{\{x\in I_{\varepsilon}: \phi_{\varepsilon}\geq \lambda\}}\frac{1}{\phi^2_{\varepsilon}}\leq \frac{\L^1(I_{\varepsilon})}{\lambda^2}\qquad\mbox{ and }\qquad\int_{\{x\in I_{\varepsilon}: \phi_{\varepsilon}< \lambda\}}\frac{1}{\phi^2_{\varepsilon}}\leq \frac{1}{(1-\lambda)^2}\frac{2{\varepsilon}}{\eta^2}{\left}(\int_{I_{\varepsilon}}\frac{(1-\phi_{\varepsilon})^2}{2{\varepsilon}}\dx{\right}).$$ Applying both of them in  we obtain $$\begin{aligned}
\nonumber
\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon},I_{\varepsilon})&\geq  h_{\varepsilon}(\phi_{\varepsilon})\\ \nonumber
&\geq\frac{2(D-d)^2}{\frac{{\varepsilon}\L^1(I_{\varepsilon})}{\lambda^2}+\frac{1}{(1-\lambda)^2}\frac{2{\varepsilon}^2}{\eta^2}{\left}(\int_{I\e}\frac{(1-\phi\e)^2}{2{\varepsilon}}\dx{\right})}+ \int_{I\e}{\left}(\frac{(1-\phi\e)^2}{2{\varepsilon}}{\right})\dx \\
&\geq 2(1-\lambda)\frac{\eta}{{\varepsilon}}(D-d)-(1-\lambda)^2\frac{\eta^2}{2{\varepsilon}}\frac{\L^1(I\e)}{\lambda^2} \label{dis2:liminf}\end{aligned}$$ where the latter is obtained by minimizing the function: $$t \mapsto \frac{2(D-d)^2}{\frac{{\varepsilon}\L^1(I_{\varepsilon})}{\lambda^2}+\frac{1}{(1-\lambda)^2}\frac{2{\varepsilon}^2}{\eta^2}t}+ t.$$ Therefore we can pass to the limit in  and obtain: $$\liminf_{{\varepsilon}{\downarrow}0}\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon},I_{\varepsilon})\geq(1-\lambda)\a\,2(D-d).$$ Sending $\lambda$ and $d$ to $0$ and recalling the estimation in  we get $$\label{eq:liminf1d}
\liminf_{{\varepsilon}{\downarrow}0}\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon},(y_1,y_2))\, \geq\, 1+\a\,2D \, =\, 1+\a|u(0^+)-u(0^-)|.$$ *Step 3.* Using Fubini’s decomposition we can rewrite the energy obtaining for every $\xi\in \S^{d-1}$, $$ \LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)\,\geq\,\int_{\Pi_\xi}{\left}( \int_{O^\xi_y}\frac{1}{2{\varepsilon}}(\phi^2_{\varepsilon})^\xi_y |(u'_{\varepsilon})^\xi_y|^2  + \frac{{\varepsilon}}{2}|(\phi'_{\varepsilon})^\xi_y |^2  + \frac{(1-(\phi_{\varepsilon})^\xi_y )^2}{2{\varepsilon}} \dt{\right})\dH^{d-1}(y).$$ Since $\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)$ is bounded, for $\H^{d-1}$ almost every $y\in \Pi_\xi$, the inner integral in the latter is also bounded, furthermore it corresponds to the functional on the one dimensional slice $O^\xi_y$ studied in the previous step evaluated on the couple $((u_{\varepsilon})^\xi_y,(\phi_{\varepsilon})^\xi_y)$. Taking the $\liminf$ of the above quantity, using , we get by Fatou’s lemma that for any $\xi\in \S^{d-1}$ and $\H^{d-1}$ almost every $y\in {\Omega}_\xi$ $$\int_{\Pi_\xi}\sum_{m_i\in(J_{u})^\xi_y}{\left}[1+\a|u^\xi_y(m_i^+)-u^\xi_y(m_i^-)| {\right}]\mathrm{d}\H^{d-1}(y)\leq \liminf_{{\varepsilon}{\downarrow}0} \LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O).$$ Therefore in force of Theorem \[teo:braides\] we have $u\in SBV(O)$. Moreover, since $(u')^\xi_y=0$ on each slice, we have $u\in \P(O)$. Applying identity  we get $$\label{eq:liminfxi} 
\liminf_{{\varepsilon}{\rightarrow}0}\LF_{\varepsilon}(u_{\varepsilon}, \phi_{\varepsilon};O) \geq 