Calculus of Variations and Geometric Measure Theory

A. Marchese - B. Wirth

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

created by marchese on 19 Oct 2018
modified on 02 Aug 2019


Published Paper

Inserted: 19 oct 2018
Last Updated: 2 aug 2019

Journal: J. Math. Anal. Appl.
Year: 2018


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.