Article ID Journal Published Year Pages File Type
5024539 Nonlinear Analysis: Theory, Methods & Applications 2017 15 Pages PDF
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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