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

## TL;DR

This paper develops a variational approximation for the Euclidean Steiner tree and Gilbert--Steiner problems, using Ginzburg--Landau energies to analyze 1-dimensional connected sets in higher-dimensional Euclidean spaces.

## Contribution

It extends previous planar analysis to higher dimensions by establishing a Gamma-convergence result for variational approximations of these geometric problems.

## Key findings

- Established Gamma-convergence of Ginzburg--Landau type energies for n ≥ 3
- Provided a new variational framework for 1D connected sets in higher dimensions
- Extended planar results to higher-dimensional Euclidean spaces

## 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 $n \geq 3$.