approximation to the Steiner Minimal Tree problem ([@Gilb_Poll], [@Ambr_Tilli] and [@Paol_Step]) by means of analogous techniques.

Here, we consider variational approximations of some energies of the form ${\mathcal{E}}_f$ through a family of functionals modeled on the Ambrosio-Tortorelli functional [@Amb_Tort1]. For being more precise, we need to introduce some material. Let $\rho\in C^\infty_c({\mathbf{R}}^2,{\mathbf{R}}_+)$ be a classical radial mollifier with $\operatorname{supp}\rho\subset B_1(0)$ and $\int \rho=1$. For ${\varepsilon}\in (0,1]$, we set $\rho_{\varepsilon}(x)={\varepsilon}^{-2}\rho({\varepsilon}^{-1}x)$ and we define the space $V_{\varepsilon}({\Omega})$ of square integrable vector fields whose weak divergence satisfy the constraint $$\label{eq:divconstraint2}
{\nabla}\cdot\s_{\varepsilon}\, =\, {\left}(N\delta_{x_0} - \sum_{j=1}^N \delta_{x_j} {\right})\ast \rho_{\varepsilon}.$$ For $\eta=\eta({\varepsilon})>0$, we note $$\label{eq:Weps}
W_{\varepsilon}({\Omega})\,=\,{\left}\{\phi\in H^1({\Omega})\,:\, \eta \leq\phi\leq 1\mbox{ in }{\Omega},\ \phi\equiv 1\mbox{ on }\de{\Omega}{\right}\}.$$ Then we define the energy ${\mathcal{F}}_{\varepsilon}:\M({\Omega},{\mathbf{R}}^2)\times L^1({\Omega}){\rightarrow}[0,+{\infty}]$ as $$\label{eq:functional}
{\mathcal{F}}_{\varepsilon}(\s,\phi):=
  \begin{dcases}
    \;\int_{{\Omega}} \frac{1}{2{\varepsilon}}\phi^2 |\s|^2 \dx+ \int_{{\Omega}} \frac{{\varepsilon}}{2}|{\nabla}\phi|^2  + \frac{(1-\phi)^2}{2{\varepsilon}} \dx, & \text{if } (\s,\phi)\in V_{\varepsilon}({\Omega})\times W_{\varepsilon}({\Omega}),\\
    \quad+{\infty}& \text{in the other cases.}
  \end{dcases}$$ The first integral in the definition of the energy will be refered to as the “constraint component” while the second integral will be regarded as the “Modica-Mortola component”. Let us briefly describe the qualitative properties of the associated minimization problem. First notice that the constraint  enforces $\s$ to be non zero on a set connecting $S$. Next, 