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On the structure of finite perimeter sets in step 2 Carnot groups.(English)Zbl 1064.49033

Given a step 2 Carnot group \(G\), it is possible to define functions with bounded \(G\)-variation and sets of finite \(G\)-perimeter using the notion of intrinsic divergence. The main purpose of the authors is to investigate the structure properties of sets with finite perimeter. In order to do this, they give the notion of \(G\)-regular sets as level sets of functions regular with respect to the differentiable structure of \(G\). With a notion of intrinsic regular sets, it is possible to define \(G\)-rectifiability. Using this notion, the authors prove that the perimeter measure of a set \(E\) with finite \(G\)-perimeter is concentrated on a \(G\)-rectifiable set, the \(G\)-reduced boundary of \(E\). They prove also a representation formula for the perimeter measure and a divergence theorem, extending the classical De Giorgi results in this framework. The proof relies on a blow-up technique on points of reduced boundary, showing that the blow-up converges to a subgroup. This phenomenon fails to happen in Carnot groups with step greater then 2, as shown by the Engel group. In the last section, the authors prove also that codimension one Euclidean regular sets are \(G\)-rectifiable, but the converse is false as shown by a counterexample.

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
22E25 Nilpotent and solvable Lie groups
53C17 Sub-Riemannian geometry
49N60 Regularity of solutions in optimal control

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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