Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Carnot groups; perimeter; rectifiability; divergence theorem
Carnot groups; perimeter; rectifiability; divergence theorem
Summary:
We study finite perimeter sets in step 2 Carnot groups. In this way we extend the classical De Giorgi’s theory, developed in Euclidean spaces by De Giorgi, as well as its generalization, considered by the authors, in Heisenberg groups. A structure theorem for sets of finite perimeter and consequently a divergence theorem are obtained. Full proofs of these results, comments and an exhaustive bibliography can be found in our preprint (2001).
References:
[1] L. Ambrosio: Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces. Adv. Math. 159 (2001), 51–67. DOI 10.1006/aima.2000.1963 | MR 1823840 | Zbl 1002.28004
[2] E. De Giorgi: Su una teoria generale della misura $(r-1)$-dimensionale in uno spazio ad $r$ dimensioni. Ann. Mat. Pura Appl. 36 (1954), 191–213. DOI 10.1007/BF02412838 | MR 0062214 | Zbl 0055.28504
[3] E. De Giorgi: Nuovi teoremi relativi alle misure $(r-1)$-dimensionali in uno spazio ad $r$ dimensioni. Ricerche Mat. 4 (1955), 95–113. MR 0074499 | Zbl 0066.29903
[4] H. Federer: Geometric Measure Theory. Springer, 1969. MR 0257325 | Zbl 0176.00801
[5] G. B.Folland, E. M. Stein: Hardy Spaces on Homogeneous Groups. Princeton University Press, 1982. MR 0657581 | Zbl 0508.42025
[6] B. Franchi, R. Serapioni, F. Serra Cassano: Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields. Houston J. Math. 22 (1996), 859–889. MR 1437714
[7] B. Franchi, R. Serapioni, F. Serra Cassano: Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321 (2001), 479–531. DOI 10.1007/s002080100228 | MR 1871966
[8] B. Franchi, R. Serapioni, F. Serra Cassano: Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups. Preprint (2001). MR 2032504
[9] B. Franchi, R. Serapioni, F. Serra Cassano: On the structure of finite perimeter sets in step 2 Carnot groups. Preprint (2001). MR 1984849
[10] N. Garofalo, D. M. Nhieu: Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math. 49 (1996), 1081–1144. DOI 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A | MR 1404326
[11] M. Gromov: Metric Structures for Riemannian and non Riemannian Spaces. vol. 152, Progress in Mathematics, Birkhauser, Boston, 1999. MR 1699320 | Zbl 0953.53002
[12] J. Heinonen: Calculus on Carnot groups. Ber., Univ. Jyväskylä 68 (1995), 1–31. MR 1351042 | Zbl 0863.22009
[13] A. Korányi, H. M. Reimann: Foundation for the theory of quasiconformal mappings on the Heisenberg group. Adv. Math. 111 (1995), 1–87. DOI 10.1006/aima.1995.1017
[14] J. Mitchell: On Carnot-Carathéodory metrics. J. Differ. Geom. 21 (1985), 35–45. DOI 10.4310/jdg/1214439462 | MR 0806700 | Zbl 0554.53023
[15] R. Monti, F. Serra Cassano: Surface measures in Carnot-Carathéodory spaces. Calc. Var. Partial Differ. Equ (to appear). MR 1865002
[16] P. Pansu: Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math. 129 (1989), 1–60. DOI 10.2307/1971484 | MR 0979599 | Zbl 0678.53042
[17] E. Sawyer, R. L. Wheeden: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Amer. J. Math. 114 (1992), 813–874. DOI 10.2307/2374799 | MR 1175693
Search
Partner of
