[Régularité des ensembles à normale intrinsèque constante dans une classe de groupes de Carnot]
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.
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, Tome 64 (2014) no. 2, pp. 429-455. doi : 10.5802/aif.2853. https://aif.centre-mersenne.org/articles/10.5802/aif.2853/
[1] Rectifiability of Sets of Finite Perimeter in Carnot Groups: Existence of a Tangent Hyperplane, J. Geom. Anal., Volume 19 (2009), pp. 509-540 | DOI | MR | Zbl
[2] Stratified Lie Groups and Potential Theory for their Sub-Laplacians, Springer, New York, 2007 | MR | Zbl
[3] Nuovi teoremi relativi alle misure -dimensionali in uno spazio ad dimensioni, Ricerche Mat., Volume 4 (1955), pp. 95-113 | MR | Zbl
[4] Elementary matrix theory, Courier Dover Publications, New York, 1980 | MR | Zbl
[5] Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., Volume 13 (1975), pp. 161-207 | DOI | MR | Zbl
[6] Hardy spaces on homogeneous groups, Princeton University Press, Princeton, 1982 | MR | Zbl
[7] Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math., Volume 22 (1996) no. 4, pp. 859-889 | MR | Zbl
[8] On the Structure of Finite Perimeter Sets in Step 2 Carnot Groups, J. Geom. Anal., Volume 13 (2003), pp. 421-466 | DOI | MR | Zbl
[9] Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups, Comm. Anal. Geom., Volume 11 (2003) no. 5, pp. 909-944 | DOI | MR | Zbl
[10] Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math., Volume 49 (1996), pp. 1081-1144 | DOI | MR | Zbl
[11] Foundations of Lie Theory and Lie Transformation Groups, Springer, Berlin, 1997 | MR | Zbl
[12] Foundation for the theory of quasiconformal mappings on the Heisenberg group, Advances in Mathematics, Volume 111 (1995), pp. 1-87 | DOI | MR | Zbl
[13] Characteristic points, rectifiability and perimeter measure on stratified groups, J. Eur. Math. Soc., Volume 8 (2006) no. 4, pp. 585-609 | DOI | MR | Zbl
[14] On Carnot-Carathéodory metrics, J. Differ. Geom., Volume 21 (1985), pp. 35-45 | MR | Zbl
[15] A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, 91, AMS, Providence RI, 2002 | MR | Zbl
[16] Balls and metrics defined by vector fields I: Basic properties, Acta Mathematica, Volume 155 (1985) no. 1, pp. 103-147 | DOI | MR | Zbl
[17] Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math., Volume 129 (1989), pp. 1-60 | DOI | MR | Zbl
[18] Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992 | MR | Zbl
Cité par Sources :
