On the lower semicontinuous envelope of functionals defined on polyhedral chains. (November 2017)

• Record Type: Journal Article
• Authors: Colombo, Maria; De Rosa, Antonio; Marchese, Andrea; Stuvard, Salvatore
• Is Part Of: Nonlinear analysis. Volume 163(2017)
• Page Start: 201
• Page End: 215
• Publication Date: 2017-11
• Subjects: 49Q15 -- 49J45; Rectifiable currents -- H-mass -- Polyhedral approximation -- Relaxation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2017.08.002
• Languages: English
• ISSNs: 0362-546X

Abstract

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 Φ H the functional induced by H on polyhedral m -chains, namely Φ H ( P ) ≔ ∑ i = 1 N H ( θ i ) H m ( σ i ), for every P = ∑ i = 1 N θ i 〚 σ i 〛 ∈ P m ( R n ), we prove that the lower semicontinuous envelope of Φ H coincides on rectifiable m -currents with the H -mass M H ( R ) ≔ ∫ E H ( θ ( x ) ) d H m ( x ) for every R = 〚 E, τ, θ 〛 ∈ R m ( R n ) .