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Anisotropic Surface Measures as Limits of Volume Fractions

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Current Research in Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 135))

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.

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Notes

  1. 1.

    Without loss of generality, we can always assume 0 ∈ D ⊂ C.

  2. 2.

    If ν 2 = 0, the length is 2εb.

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Acknowledgements

The authors would like to thank Emanuele Paolini for fruitful discussions concerning the examples of anisotropic coverings (Sect. 4.3).

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Authors and Affiliations

  1. Scuola Normale Superiore, Pisa, Italy

    Luigi Ambrosio & Giovanni E. Comi

Authors
  1. Luigi Ambrosio
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  2. Giovanni E. Comi
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Corresponding author

Correspondence to Luigi Ambrosio .

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Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Ambrosio, L., Comi, G.E. (2018). Anisotropic Surface Measures as Limits of Volume Fractions. In: Rassias, T. (eds) Current Research in Nonlinear Analysis. Springer Optimization and Its Applications, vol 135. Springer, Cham. https://doi.org/10.1007/978-3-319-89800-1_1

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