# On the lower semicontinuous envelope of functionals defined on polyhedral chains

## TL;DR

This paper derives an explicit formula for the lower semicontinuous envelope of certain functionals on polyhedral chains, showing it coincides with the H-mass on rectifiable currents, advancing the understanding of variational problems in geometric measure theory.

## Contribution

It provides a precise characterization of the lower semicontinuous envelope of functionals on polyhedral chains, linking it explicitly to the H-mass for rectifiable currents.

## Key findings

- The lower semicontinuous envelope matches the H-mass on rectifiable currents.
- An explicit formula for the envelope is established.
- The results enhance the understanding of variational functionals in geometric measure theory.

## 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 $H \colon \mathbb{R} \to \left[ 0,\infty \right)$ an even, subadditive, and lower semicontinuous function with $H(0)=0$, and by $\Phi_H$ the functional induced by $H$ on polyhedral $m$-chains, namely \[ \Phi_{H}(P) := \sum_{i=1}^{N} H(\theta_{i}) \mathcal{H}^{m}(\sigma_{i}), \quad\mbox{for every }P=\sum_{i=1}^{N} \theta_{i} [[ \sigma_{i} ]] \in\mathbf{P}_m(\mathbb{R}^n), \] we prove that the lower semicontinuous envelope of $\Phi_H$ coincides on rectifiable $m$-currents with the $H$-mass \[ \mathbb{M}_{H}(R) := \int_E H(\theta(x)) \, d\mathcal{H}^m(x) \quad \mbox{ for every } R= [[ E,\tau,\theta ]] \in \mathbf{R}_{m}(\mathbb{R}^{n}). \]