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Anisotropic Surface Measures as Limits of Volume Fractions

In: Current Research in Nonlinear Analysis

Author

Listed:
  • Luigi Ambrosio

    (Scuola Normale Superiore)

  • Giovanni E. Comi

    (Scuola Normale Superiore)

Abstract

In this paper we consider the new characterization of the perimeter of a measurable set in ℝ n $$\mathbb {R}^{n}$$ 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.

Suggested Citation

  • Luigi Ambrosio & Giovanni E. Comi, 2018. "Anisotropic Surface Measures as Limits of Volume Fractions," Springer Optimization and Its Applications, in: Themistocles M. Rassias (ed.), Current Research in Nonlinear Analysis, pages 1-32, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-89800-1_1
    DOI: 10.1007/978-3-319-89800-1_1
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