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 by stuvard on 04 Sep 2017

[BibTeX]

Published Paper

Inserted: 24 feb 2017
Last Updated: 4 sep 2017

Journal: Nonlinear Analysis
Volume: 163C
Pages: 201-215
Year: 2017
Doi: 10.1016/j.na.2017.08.002

ArXiv: 1703.01938 PDF
Links: Article Link

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

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


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