Phase field models for two-dimensional branched transportation problems

TL;DR

This paper investigates phase field models to approximate complex transportation network functionals in two dimensions, providing existence, characterization, and explicit approximation formulas for solutions.

Contribution

It introduces a novel approach to approximate nonconvex transportation network functionals using phase field models, with proven existence and explicit solution formulas.

Findings

Existence of solutions for the phase field approximation.

Characterization of solutions via linear deconvolution.

Explicit formulas for arbitrary approximation accuracy.

Abstract

We analyse the following inverse problem. Given a nonconvex functional (from a specific, but quite general class) of normal, codimension-1 currents (which in two spatial dimensions can be interpreted as transportation networks), find the potential of a phase field energy which approximates the given functional. We prove existence of a solution as well as its characterization via a linear deconvolution problem. We also provide an explicit formula that allows to approximate the solution arbitrarily well in the supremum norm.