Calculus of Variations and Geometric Measure Theory

R. Korte - P. Lahti - N. Shanmugalingam

Semmes family of curves and a characterization of functions of bounded variation in terms of curves

created by shanmugal on 02 Sep 2013
modified on 24 Jul 2015


Accepted Paper

Inserted: 2 sep 2013
Last Updated: 24 jul 2015

Journal: Calc. Var. PDE
Year: 2015


A Federer-type characterization of Euclidean sets of finite perimeter states that a set has finite perimeter if and only if its essential boundary (measure-theoretic boundary) has finite (n-1)-dimensional Hausdorff measure. Here we show that if a metric measure space, equipped with a doubling measure, supports a Semmes family of curves satisfying certain geometric conditions, then a measurable set has finite perimeter if and only if the co-dimension 1 Hausdorff measure of its measure-theoretic boundary is finite. The fact that if a measurable set has finite perimeter then its measure-theoretic boundary has finite co-dimension 1 Hausdorff measure is due to Ambrosio. To prove the converse, in this paper we first show a characterization of BV functions in terms of "BV along Semmes family of curves".