believe our approach is slightly simpler and numerically easier to implement. We show that we can modify classical approximations for free discontinuity problems [@Mod_Mort; @Amb_Tort1; @Iur; @ContiFocardiIurlano] to address our specific problem, where the limiting energy is concentrated only on a singular one-dimensional network (and roughly measures its length). Numerical results illustrate the behaviour of these elliptic approximations. In this first study, we limit ourselves to the two-dimensional case, as in that case lines can locally be seen as discontinuities of piecewise constant functions, so that our construction is deriving in a quite simple ways from the above mentioned previous works on free discontinuity problems. Higher dimension is more challenging, from the topological point of view; an extension of this approach is currently in preparation.

We now introduce precisely our mathematical framework. Let ${\Omega}\subset {\mathbf{R}}^2$ be a convex, bounded open set. We consider measures $\s\in \M({\overline}{{\Omega}},{\mathbf{R}}^2)$ that write $$\s=\t\xi\cdot\H^1{ \!\!\mbox{{ \Large$\llcorner$}} }M,$$ where $M$ is a $1$-dimensional rectifiable set orientated by a Borel measurable mapping $\xi:M{\rightarrow}\S^1$ and $\t:M{\rightarrow}{\mathbf{R}}_+$ is a Borel measurable function representing the multiplicity. Such measure is called a rectifiable measure. We follow the notation of [@OS2011] and write $\s=U(M,\t,\xi)$. Given a cost function $f\in C({\mathbf{R}}_+,{\mathbf{R}}_+)$, we introduce 