Anisotropic Surface Measures as Limits of Volume Fractions

Luigi Ambrosio () and Giovanni E. Comi ()
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Luigi Ambrosio: Scuola Normale Superiore
Giovanni E. Comi: Scuola Normale Superiore

A chapter in Current Research in Nonlinear Analysis, 2018, pp 1-32 from Springer

Abstract: 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.

Date: 2018
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DOI: 10.1007/978-3-319-89800-1_1

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