Calculus of Variations and Geometric Measure Theory
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L. Ambrosio - G. E. Comi

Anisotropic surface measures as limits of volume fractions

created by comi on 13 Oct 2016

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Submitted Paper

Inserted: 13 oct 2016
Last Updated: 13 oct 2016

Year: 2016

Abstract:

In this paper we consider the new characterization of the perimeter of a measurable set in $\mathbb{R}^{n}$ recently studied by Ambrosio, Bourgain, Brezis and Figalli. We modify their approach by using, instead of cubes, covering families made by translations of a given open bounded set with Lipschitz boundary. We show that the new functionals converge to an anisotropic surface measure, which is indeed a multiple of the perimeter if we allow for isotropic coverings (e.g. balls or arbitrary rotations of the given set). This result underlines that the particular geometry of the covering sets is not essential.

Keywords: Sets of finite perimeter, anisotropic perimeter, BMO-type norms, sphere packing problem


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