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research article
On the lower semicontinuous envelope of functionals defined on polyhedral chains
November 1, 2017
In this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains. More precisely, denoting by H:R→[0,∞) an even, subadditive, and lower semicontinuous function with H(0)=0, and by ΦH the functional induced by H on polyhedral m-chains, namely ΦH(P)≔∑i=1NH(θi)Hm(σi),for every P=∑i=1Nθi〚σi〛∈Pm(Rn),we prove that the lower semicontinuous envelope of ΦH coincides on rectifiable m-currents with the H-mass MH(R)≔∫EH(θ(x))dHm(x) for every R=〚E,τ,θ〛∈Rm(Rn).
Type
research article
Authors
Publication date
2017-11-01
Published in
Volume
163
Start page
201
End page
215
Peer reviewed
REVIEWED
EPFL units
Available on Infoscience
February 13, 2020
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