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.
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.
Keywords: Carnot groups, intrinsic perimeter, intrinsic rectifiability
Mot clés : Groupes de Carnot, périmètre intrinsèque, rectifiabilité intrinsèque
Marchi, Marco 1
@article{AIF_2014__64_2_429_0, author = {Marchi, Marco}, title = {Regularity of sets with constant intrinsic normal in a class of {Carnot} groups}, journal = {Annales de l'Institut Fourier}, pages = {429--455}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {64}, number = {2}, year = {2014}, doi = {10.5802/aif.2853}, mrnumber = {3330910}, zbl = {06387280}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2853/} }
TY - JOUR AU - Marchi, Marco TI - Regularity of sets with constant intrinsic normal in a class of Carnot groups JO - Annales de l'Institut Fourier PY - 2014 SP - 429 EP - 455 VL - 64 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2853/ DO - 10.5802/aif.2853 LA - en ID - AIF_2014__64_2_429_0 ER -
%0 Journal Article %A Marchi, Marco %T Regularity of sets with constant intrinsic normal in a class of Carnot groups %J Annales de l'Institut Fourier %D 2014 %P 429-455 %V 64 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2853/ %R 10.5802/aif.2853 %G en %F AIF_2014__64_2_429_0
Marchi, Marco. Regularity of sets with constant intrinsic normal in a class of Carnot groups. Annales de l'Institut Fourier, Volume 64 (2014) no. 2, pp. 429-455. doi : 10.5802/aif.2853. https://aif.centre-mersenne.org/articles/10.5802/aif.2853/
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