Calculus of Variations and Geometric Measure Theory

L. Ambrosio

Some Fine Properties of Sets of Finite Perimeter in Ahlfors Regular Metric Measure Spaces

created on 22 Feb 2000
modified on 19 Dec 2001


Published Paper

Inserted: 22 feb 2000
Last Updated: 19 dec 2001

Journal: Advances in Mathematics
Number: 159
Pages: 51-67
Year: 2001


We prove that for any set of finite perimeter in an Ahlfors $k$-regular metric space admitting a weak $(1,1)$-Poincaré inequality the perimeter measure is concentrated on the essential boundary of the set, i.e. the set of points where neither $E$ nor its complement have zero density. Moreover, the perimeter measure is absolutely continuous with respect to the Hausdorff measure $\cal H^{k-1}$. We obtain also density lower bounds on volume and perimeter which lead to the (asymptotic) doubling property of the perimeter measure: this property is of interest in connection with the rectifiability problem in the Heisenberg group and, more generally, in Carnot-Carath{é}odory groups.

Keywords: perimeter, metric spaces