A Modica-Mortola approximation for branched transport

TL;DR

This paper introduces a Modica-Mortola type approximation for the M^lpha energy in branched transport, proving convergence of elliptic energies to handle singular measures with prescribed divergence.

Contribution

It develops a novel approximation method for branched transport energies using elliptic functionals inspired by the Modica-Mortola approach, enabling analysis of singular measures.

Findings

Proves convergence of elliptic energies to the branched transport energy.

Provides a new regularization technique for singular measures in transport problems.

Establishes a theoretical foundation for numerical approximation of branched transport energies.

Abstract

The M^\alpha energy which is usually minimized in branched transport problems among singular 1-dimensional rectifiable vector measures with prescribed divergence is approximated (and convergence is proved) by means of a sequence of elliptic energies, defined on more regular vector fields. The procedure recalls the Modica-Mortola one for approximating the perimeter, and the double-well potential is replaced by a concave power.