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 / Colombo, M.; De Rosa, A.; Marchese, A.; Stuvard, S.. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 163:(2017), pp. 201-215. [10.1016/j.na.2017.08.002]

On the lower semicontinuous envelope of functionals defined on polyhedral chains

Marchese A.;
2017-01-01

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).
2017
Colombo, M.; De Rosa, A.; Marchese, A.; Stuvard, S.
On the lower semicontinuous envelope of functionals defined on polyhedral chains / Colombo, M.; De Rosa, A.; Marchese, A.; Stuvard, S.. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 163:(2017), pp. 201-215. [10.1016/j.na.2017.08.002]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/265901
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