Calculus of Variations and Geometric Measure Theory

L. Ambrosio - G. E. Comi

Anisotropic surface measures as limits of volume fractions

created by comi on 13 Oct 2016
modified on 19 Dec 2018


Published Paper

Inserted: 13 oct 2016
Last Updated: 19 dec 2018

Journal: Current Research in Nonlinear Analysis: In Honor of Haim Brezis and Louis Nirenberg
Pages: 1-32
Year: 2018
Doi: 10.1007/978-3-319-89800-1_1


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 connected 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