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 $n\ge 3$.

17 pages, 1 figure

References in corpus (2)

• Approximation of length minimization problems among compact connected sets
• The Steiner tree problem revisited through rectifiable G-currents