*Advances in Calculus of Variations*

**Inserted:** 10 feb 2017

**Last Updated:** 6 jul 2017

**Journal:** Advances in Calculus of Variations

**Year:** 2017

**Doi:** 10.1515/acv-2016-0034

**Notes:**

24 pages, 8 figures

**Abstract:**

In this paper we consider the branched transportation problem in 2D associated with a cost per unit length of the form 1+αm where m denotes the amount of transported mass and α>0 is a fixed parameter (notice that the limit case α=0 corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ({Fϵ}ϵ>0) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of {Fϵ} as ϵ↓0. Our functionals are modeled on the Ambrosio-Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method

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