Federer’s characterization of sets of finite perimeter in metric spaces

Lahti, Panu

doi:10.2140/apde.2020.13.1501

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Federer's characterization of sets of finite perimeter in metric spaces

Panu Lahti

Vol. 13 (2020), No. 5, 1501–1519

DOI: 10.2140/apde.2020.13.1501

Abstract

Federer’s characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension 1. In complete metric spaces that are equipped with a doubling measure and support a Poincaré inequality, the “only if” direction was shown by Ambrosio (2002). By applying fine potential theory in the case $p = 1$ , we prove that the “if” direction holds as well.

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Keywords

metric measure space, set of finite perimeter, Federer's characterization, measure-theoretic boundary, codimension-1 Hausdorff measure, fine topology

Mathematical Subject Classification 2010

Primary: 30L99, 31E05, 26B30

Milestones

Received: 27 April 2018

Revised: 3 January 2019

Accepted: 12 May 2019

Published: 27 July 2020

Authors

Panu Lahti
Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

Vol. 13, No. 5, 2020