Energy minimizing maps with prescribed singularities and Gilbert-Steiner optimal networks

TL;DR

This paper explores the connection between energy-minimizing maps with singularities and optimal network problems like Steiner trees, establishing an equivalence through a homological Plateau problem framework.

Contribution

It generalizes previous work by linking variational problems for maps with singularities to optimal network problems via a homological approach.

Findings

Equivalence between energy minimization and optimal network problems.

Interpretation of branched transport as a homological Plateau problem.

Generalization of prior results by Brezis, Coron, and Lieb.

Abstract

We investigate the relation between energy minimizing maps valued into spheres having topological singularities at given points and optimal networks connecting them (e.g. Steiner trees, Gilbert-Steiner irrigation networks). We show the equivalence of the corresponding variational problems, interpreting in particular the branched optimal transport problem as a homological Plateau problem for rectifiable currents with values in a suitable normed group. This generalizes the pioneering work by Brezis, Coron and Lieb [10].