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Rectifiability and perimeter in the Heisenberg group. (English) Zbl 1057.49032

The authors investigate the properties of \({\mathcal H}\)–Caccioppoli sets in the \(n\)–dimensional Heisenberg group \({\mathcal H}^n\). In particular, they are interested in the study of rectifiability results for the reduced boundary of \({\mathcal H}\)–Caccioppoli sets. In order to do that, some technical tools are introduced. First of all, a good and intrinsic notion of regular sets is needed; for this reason, the authors give the definition of \({\mathcal H}\)–regular functions and define regular sets as the regular level sets of such functions. In order to deal with this definition, a very important role is played by the Implicit Function Theorem proved in this paper for \({\mathcal H}\)–regular functions. Once these basic tools are introduced, the authors are able to prove that an \({\mathcal H}\)–Caccioppoli set is \({\mathcal H}\)–rectifiable in the sense that almost all of the reduced boundary of an \({\mathcal H}\)–Caccioppoli set is contained in a countable union of \({\mathcal H}\)–regular sets. Moreover, the following representation formula for the perimeter measure is given \[ | \partial E| _{\mathcal H}=\frac{2\omega_{2n-1}}{\omega_{2n+1}}{\mathcal S}^{Q-1}_d \lfloor \partial ^* _{{\mathcal H}}E, \] where \(| \partial E| _{\mathcal H}\) is the \({\mathcal H}\)–perimeter, \(Q\) is the homogeneous dimension of \({\mathcal H}^n\), \(d\) is the left-invariant metric induced by the homogeneous norm \[ \| P\|_\infty =\max\{ | z| ,| t| ^{1/2}\},\quad P=[z,t], z\in {\mathbb C}^n,t\in {\mathbb R}, \] \({\mathcal S}^{Q-1}_d\) is the spherical Hausdorff measure and \(\partial ^*_{\mathcal H}E\) is the essential boundary of \(E\).

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q05 Minimal surfaces and optimization
22E25 Nilpotent and solvable Lie groups
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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