Variational approximation of functionals defined on 1-dimensional connected sets: the planar case

TL;DR

This paper develops a variational approximation framework for 1D connected set problems like Steiner trees and irrigation in the plane, using $ ext{Gamma}$-convergence, convex relaxation, and numerical methods, with extensions to higher dimensions.

Contribution

It introduces a novel variational approximation and convex relaxation for 1D connected set problems, along with numerical implementations and generalizations to higher dimensions.

Findings

Established $ ext{Gamma}$-convergence of the approximation

Developed a convex relaxation suitable for numerical methods

Results extend to higher-dimensional Euclidean spaces

Abstract

In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a $\Gamma$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to $n$-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [11].