On the lower semicontinuous envelope of functionals defined on polyhedral chains

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 ΦH the functional induced by H on polyhedral m-chains, namelywe prove that the lower semicontinuous envelope of ΦH coincides on rectifiable m-currents with the H-mass) dH m (x) for every R = E, τ, θ ∈ Rm(R n ).

Strong approximation inh-mass of rectifiable currents under homological constraint

“…Applying the theorem with R = U 1 as above and using the subadditivity of M h , we obtain the desired result. Unfortunately, (1.10) is not stated in [6]. However, in the proof of [6, Proposition 2.6] the currents U 1 and V 1 obtained in (1.8) are rectifiable by construction and with obvious modifications 2 we can assume that U 1 and V 1 satisfy the estimate (1.10).…”

“…In order to establish (1.13) we should prove that M h is lower semicontinuous with respect to the flat norm topology. This is out of the scope of the present note but we believe that this can be done with a method based on slicing as in [7,6].…”

“…In the recent paper [6], it is established that under conditions in (1.2), M h is lower semi-continuous on R k (R n ). The result is more precise.…”

“…We note P k (R n ) ⊂ R k (R n ) the space of k-polyhedral currents. In [6], the authors introduce the lower semicontinuous envelope of M h restricted to P k (R n ) with respect to the flat convergence:…”

“…h is non-decreasing on (0, +∞) with lim θ↓0 h(θ)/θ = +∞, (1.4) it is established that Φ h ≡ +∞ on F k (R n ) \ R k (R n ) (see [6,Prop. 2.7]).…”