Calculus of Variations and Geometric Measure Theory
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M. Colombo - A. De Rosa - A. Marchese - S. Stuvard

On the lower semicontinuous envelope of functionals defined on polyhedral chains

created by marchese on 24 Feb 2017
modified on 06 Mar 2017

[BibTeX]

Preprint

Inserted: 24 feb 2017
Last Updated: 6 mar 2017

Year: 2017

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 $H:\mathbb{R}\to[0,\infty)$ 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 \[\mathbb{M}_{H}(R) := \int_E H(\theta(x)) \, d\mathcal{H}^m(x) \quad \mbox{ for every } R=[[ E,\tau,\theta]] \in \mathbf{R}_{m}(\mathbb{R}^{n}). \]

Keywords: Rectifiable currents, H-mass, Polyhedral approximation, Relaxation.


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