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 show that, up to a null set, these isoperimetric sets are open, bounded, their boundary is Ahlfors-regular and they satisfy the condition B. Furthermore, in the particular case of the Heisenberg group, we prove that any reduced isoperimetric set is a domain of isoperimetry. All these properties are satisfied with implicit constants that depend only on the dimension of the group and on the prescribed Lebesgue measure.
Isoperimetric sets on Carnot groups / Leonardi, Gian Paolo; Rigot, Severine. - In: HOUSTON JOURNAL OF MATHEMATICS. - ISSN 0362-1588. - STAMPA. - 29:3(2003), pp. 609-637.
Isoperimetric sets on Carnot groups
LEONARDI, Gian Paolo;
2003-01-01
Abstract
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 show that, up to a null set, these isoperimetric sets are open, bounded, their boundary is Ahlfors-regular and they satisfy the condition B. Furthermore, in the particular case of the Heisenberg group, we prove that any reduced isoperimetric set is a domain of isoperimetry. All these properties are satisfied with implicit constants that depend only on the dimension of the group and on the prescribed Lebesgue measure.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione