BV functions and sets of finite perimeter in sub-Riemannian manifolds
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 3, pp. 489-517.

We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms G p :T p M[0,] are given. When we consider sub-Riemannian manifolds, our definition coincides with the one given in the more general context of metric measure spaces which are doubling and support a Poincaré inequality. We focus on finite perimeter sets, i.e., sets whose characteristic function is BV, in sub-Riemannian manifolds. Under an assumption on the nilpotent approximation, we prove a blowup theorem, generalizing the one obtained for step-2 Carnot groups in [24].

@article{AIHPC_2015__32_3_489_0,
     author = {Ambrosio, L. and Ghezzi, R. and Magnani, V.},
     title = {BV functions and sets of finite perimeter in {sub-Riemannian} manifolds},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {489--517},
     publisher = {Elsevier},
     volume = {32},
     number = {3},
     year = {2015},
     doi = {10.1016/j.anihpc.2014.01.005},
     mrnumber = {3353698},
     zbl = {1320.53034},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.01.005/}
}
TY  - JOUR
AU  - Ambrosio, L.
AU  - Ghezzi, R.
AU  - Magnani, V.
TI  - BV functions and sets of finite perimeter in sub-Riemannian manifolds
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 489
EP  - 517
VL  - 32
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2014.01.005/
DO  - 10.1016/j.anihpc.2014.01.005
LA  - en
ID  - AIHPC_2015__32_3_489_0
ER  - 
%0 Journal Article
%A Ambrosio, L.
%A Ghezzi, R.
%A Magnani, V.
%T BV functions and sets of finite perimeter in sub-Riemannian manifolds
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 489-517
%V 32
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2014.01.005/
%R 10.1016/j.anihpc.2014.01.005
%G en
%F AIHPC_2015__32_3_489_0
Ambrosio, L.; Ghezzi, R.; Magnani, V. BV functions and sets of finite perimeter in sub-Riemannian manifolds. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 3, pp. 489-517. doi : 10.1016/j.anihpc.2014.01.005. http://www.numdam.org/articles/10.1016/j.anihpc.2014.01.005/

[1] A. Agrachev, D. Barilari, U. Boscain, Introduction to Riemannian and sub-Riemannian geometry, http://www.cmapx.polytechnique.fr/~barilari/Notes.php (2012) | Zbl

[2] A. Agrachev, D. Barilari, U. Boscain, On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. Partial Differ. Equ. 43 (2012), 355 -388 | MR | Zbl

[3] A. Agrachev, U. Boscain, M. Sigalotti, A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds, Discrete Contin. Dyn. Syst. 20 no. 4 (2008), 801 -822 | MR | Zbl

[4] A.A. Agrachev, U. Boscain, G. Charlot, R. Ghezzi, M. Sigalotti, Two-dimensional almost-Riemannian structures with tangency points, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27 no. 3 (2010), 793 -807 | Numdam | MR | Zbl

[5] A.A. Agrachev, Y.L. Sachkov, Control theory from the geometric viewpoint, Control Theory and Optimization, II, Encycl. Math. Sci. vol. 87 , Springer-Verlag, Berlin (2004) | MR | Zbl

[6] L. Ambrosio, Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv. Math. 159 no. 1 (2001), 51 -67 | MR | Zbl

[7] L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces, Set-Valued Anal. 10 no. 2–3 (2002), 111 -128 | MR | Zbl

[8] L. Ambrosio, M. Colombo, S. Di Marino, Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope, arXiv:1212.3779v1 (2012)

[9] L. Ambrosio, G. Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, Transport Equations and Multi-D Hyperbolic Conservation Laws, Lect. Notes Unione Mat. Ital. vol. 5 , Springer, Berlin (2008), 3 -57 | MR | Zbl

[10] L. Ambrosio, S. Di Marino, Equivalent definitions of BV space and of total variation on metric measure spaces, http://cvgmt.sns.it/paper/1860/ (2012) | Zbl

[11] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxf. Math. Monogr. , The Clarendon Press, Oxford University Press, New York (2000) | MR | Zbl

[12] L. Ambrosio, B. Kirchheim, Currents in metric spaces, Acta Math. 185 no. 1 (2000), 1 -80 | MR | Zbl

[13] L. Ambrosio, B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 no. 3 (2000), 527 -555 | MR | Zbl

[14] L. Ambrosio, B. Kleiner, E. Le Donne, Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane, J. Geom. Anal. 19 no. 3 (2009), 509 -540 | MR | Zbl

[15] A. Bellaïche, The tangent space in sub-Riemannian geometry, Sub-Riemannian Geometry, Prog. Math. vol. 144 , Birkhäuser, Basel (1996), 1 -78 | MR | Zbl

[16] W.-L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98 -105 | EuDML | JFM | MR

[17] G. Citti, M. Manfredini, Blow-up in non homogeneous lie groups and rectifiability, Houst. J. Math. 31 no. 2 (2005), 333 -353 | MR | Zbl

[18] E. De Giorgi, Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954), 191 -213 | MR | Zbl

[19] E. De Giorgi, Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio ad r dimensioni, Ric. Mat. 4 (1955), 95 -113 | MR | Zbl

[20] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math. , CRC Press, Boca Raton, FL (1992) | MR | Zbl

[21] H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. vol. 153 , Springer-Verlag, New York Inc., New York (1969) | MR | Zbl

[22] B. Franchi, R. Serapioni, F. Serra Cassano, Meyers–Serrin type theorems and relaxation of variational integrals depending on vector fields, Houst. J. Math. 22 no. 4 (1996), 859 -890 | MR | Zbl

[23] B. Franchi, R. Serapioni, F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann. 321 no. 3 (2001), 479 -531 | MR | Zbl

[24] B. Franchi, R. Serapioni, F. Serra Cassano, On the structure of finite perimeter sets in step 2 Carnot groups, J. Geom. Anal. 13 no. 3 (2003), 421 -466 | MR | Zbl

[25] N. Garofalo, D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces, Commun. Pure Appl. Math. 49 no. 10 (1996), 1081 -1144 | MR | Zbl

[26] J.-P. Gauthier, V. Zakalyukin, On the codimension one motion planning problem, J. Dyn. Control Syst. 11 no. 3 (2005), 73 -89 | MR | Zbl

[27] R.W. Goodman, Nilpotent Lie Groups: Structure and Applications to Analysis, Lect. Notes Math. vol. 562 , Springer-Verlag, Berlin (1976) | MR | Zbl

[28] M. Gromov, Carnot–Carathéodory spaces seen from within, Sub-Riemannian Geometry, Prog. Math. vol. 144 , Birkhäuser, Basel (1996), 79 -323 | MR | Zbl

[29] H. Hermes, Nilpotent and high-order approximations of vector field systems, SIAM Rev. 33 no. 2 (1991), 238 -264 | MR | Zbl

[30] D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 no. 2 (1986), 503 -523 | MR | Zbl

[31] B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Am. Math. Soc. 121 no. 1 (1994), 113 -123 | MR | Zbl

[32] E. Lanconelli, D. Morbidelli, On the Poincaré inequality for vector fields, Ark. Mat. 38 no. 2 (2000), 327 -342 | MR | Zbl

[33] M. Marchi, Rectifiability of sets of finite perimeter in a class of Carnot groups of arbitrary step, arXiv:1201.3277v1 (2012)

[34] G.A. Margulis, G.D. Mostow, Some remarks on the definition of tangent cones in a Carnot–Carathéodory space, J. Anal. Math. 80 (2000), 299 -317 | MR | Zbl

[35] P. Mattila, R. Serapioni, F. Serra Cassano, Characterizations of intrinsic rectifiability in Heisenberg groups, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 9 no. 4 (2010), 687 -723 | Numdam | MR | Zbl

[36] M. Miranda, Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl. 82 no. 8 (2003), 975 -1004 | MR | Zbl

[37] A. Nagel, E.M. Stein, S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 no. 1–2 (1985), 103 -147 | MR | Zbl

[38] P. Pansu, Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 no. 1 (1989), 1 -60 | MR | Zbl

[39] L.P. Rothschild, E.M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 no. 3–4 (1976), 247 -320 | MR | Zbl

Cité par Sources :