Regularity of sets with constant intrinsic normal in a class of Carnot groups
[Régularité des ensembles à normale intrinsèque constante dans une classe de groupes de Carnot]
Marchi, Marco
Annales de l'Institut Fourier, Tome 64 (2014), p. 429-455 / Harvested from Numdam
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Dans cette note, nous définissons une classe de groupes de Lie stratifiés de pas arbitraire (que nous appelons “groupes de type ☆” dans cet article), et nous montrons que, dans ces groupes, les ensembles à normale intrinsèque constante sont des hyperplans. En conséquence, la frontière réduite d’un ensemble de périmètre intrinsèque fini dans un groupe de type ☆ est rectifiable au sens intrinsèque (théorème de rectifiabilité de De Giorgi). Ce résultat étend un résultat précédent prouvé par Franchi, Serapioni & Serra Cassano pour les groupes de pas 2.

In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type ☆” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type ☆ is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano in step 2 groups.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/aif.2853
Classification:  28A75,  49Q15,  58C35
Mots clés: Groupes de Carnot, périmètre intrinsèque, rectifiabilité intrinsèque 
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     author = {Marchi, Marco},
     title = {Regularity of sets with constant intrinsic normal in a class of Carnot groups},
     journal = {Annales de l'Institut Fourier},
     volume = {64},
     year = {2014},
     pages = {429-455},
     doi = {10.5802/aif.2853},
     zbl = {06387280},
     mrnumber = {3330910},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2014__64_2_429_0}
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Marchi, Marco. Regularity of sets with constant intrinsic normal in a class of Carnot groups. Annales de l'Institut Fourier, Tome 64 (2014) pp. 429-455. doi : 10.5802/aif.2853. http://gdmltest.u-ga.fr/item/AIF_2014__64_2_429_0/

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