*Accepted Paper*

**Inserted:** 7 mar 2014

**Last Updated:** 10 feb 2015

**Journal:** SIAM Journal on Mathematical Analysis

**Year:** 2014

**Abstract:**

In this paper we provide an approximation à la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce a term of new type relying on a weighted geodesic distance that forces the minimizers to be connected at the limit. We apply this approach to approximate the so-called Steiner Problem, but also the average distance problem, and finally a problem relying on the $p$-compliance energy. The proof of convergence of the approximating functional, which is stated in terms of $\Gamma$-convergence relies on technical tools from geometric measure theory, as for instance a uniform lower bound for a sort of average directional Minkowski content of a family of compact connected sets.

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