In this paper we consider the new characterization of the perimeter of a measurable set in ℝn 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.
Ambrosio L., Comi G.E. (2018). Anisotropic surface measures as limits of volume fractions. Cham : Springer International Publishing [10.1007/978-3-319-89800-1_1].
Anisotropic surface measures as limits of volume fractions
Comi G. E.
2018
Abstract
In this paper we consider the new characterization of the perimeter of a measurable set in ℝn 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
