Approximation of rectifiable $1$-currents and weak-$\ast$ relaxation of the $h$-mass

TL;DR

This paper proves that rectifiable 1-currents can be approximated by polyhedral currents with controlled boundary mass, and shows the equivalence of generalized branched transport and the $h$-mass through relaxation analysis.

Contribution

It establishes a mass approximation result for rectifiable 1-currents and characterizes the relaxation of the $h$-mass, linking branched transport to $h$-mass concepts.

Findings

Approximation of rectifiable 1-currents by polyhedral currents with boundary control.

Relaxation of the $h$-mass coincides with the $h$-mass for normal currents.

Equivalence between generalized branched transport and the $h$-mass.

Abstract

Based on Smirnov's decomposition theorem we prove that every rectifiable $1$-current $T$ with finite mass $\mathbb{M}(T)$ and finite mass $\mathbb{M}(\partial T)$ of its boundary $\partial T$ can be approximated in mass by a sequence of rectifiable $1$-currents $T_n$ with polyhedral boundary $\partial T_n$ and $\mathbb{M}(\partial T_n)$ no larger than $\mathbb{M}(\partial T)$. Using this result we can compute the relaxation of the $h$-mass for polyhedral $1$-currents with respect to the joint weak-$\ast$ convergence of currents and their boundaries. We obtain that this relaxation coincides with the usual $h$-mass for normal currents. This shows that the concepts of so-called generalized branched transport and the $h$-mass are equivalent.