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).
On the lower semicontinuous envelope of functionals defined on polyhedral chains
De Rosa, Antonio;
2017
Abstract
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).
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