Calculus of Variations and Geometric Measure Theory

G. P. Leonardi - S. Rigot

Isoperimetric sets in Carnot groups

created on 18 Sep 2001
modified on 07 Jun 2002


Accepted Paper

Inserted: 18 sep 2001
Last Updated: 7 jun 2002

Journal: Houston J. Math.
Year: 2002


We prove the existence of isoperimetric sets in any Carnot group, that is, sets minimizing the intrinsic perimeter among all measurable sets with prescribed Lebesgue measure. We also give some properties of these isoperimetric sets. Namely we show that, up to a null set, they are open, bounded, their boundary is an Ahlfors-regular set of dimension $Q-1$, where $Q$ denotes the homogeneous dimension of the group, and they satisfy the condition B. In the particular case of the Heisenberg group we also prove that any reduced isoperimetric set is a domain of isoperimetry. Moreover all these properties are satisfied with implicit constants that depend only on the dimension of the group and on the prescribed Lebesgue measure.