Variational approximation of functionals defined on $1$-dimensional connected sets in $\mathbb{R}^n$

Abstract

In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert--Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in $\mathbb{R}^n$. Following the the analysis for the planar case presented in [4], we provide a variational approximation through Ginzburg--Landau type energies proving a $\Gamma$-convergence result for $n \geq 3$.