Approximation of rectifiable 1-currents and weak-⁎ relaxation of the h-mass

Based on Smirnov's decomposition theorem we prove that every rectifiable 1-current T with finite mass M(T) and finite mass M(∂T) of its boundary ∂T can be approximated in mass by a sequence of rectifiable 1-currents Tn with polyhedral boundary ∂Tn and M(∂Tn) no larger than M(∂T). Using this res...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 479; no. 2; pp. 2268 - 2283
Main Authors Marchese, Andrea, Wirth, Benedikt
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.11.2019
Subjects
Branched transport
Flat convergence
Rectifiable 1-currents
Relaxation
Branched transport
Relaxation
Rectifiable 1-currents
Flat convergence
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Summary:Based on Smirnov's decomposition theorem we prove that every rectifiable 1-current T with finite mass M(T) and finite mass M(∂T) of its boundary ∂T can be approximated in mass by a sequence of rectifiable 1-currents Tn with polyhedral boundary ∂Tn and M(∂Tn) no larger than M(∂T). Using this result we can compute the relaxation of the h-mass for polyhedral 1-currents with respect to the joint weak-⁎ 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.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2019.07.059