Calculus of Variations and Geometric Measure Theory
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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


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