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arXiv · 1810.08400

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

Andrea Marchese·Benedikt Wirth
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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.

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BibTeXRIS

Andrea Marchese, Benedikt Wirth. 2018-10-19. Approximation of rectifiable $1$-currents and weak-$\ast$ relaxation of the $h$-mass. https://arxiv.org/abs/1810.08400

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