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Generalized Schrödinger-Poisson type systems
A new class of $(H^k,1)$-rectifiable subsets of metric spaces
1. | CMAP, École Polytechnique -- Team GECO, INRIA Saclay, France |
2. | ENSTA ParisTech, UMA, and Team GECO, INRIA Saclay -- Île-de-France, Paris, France |
References:
[1] |
A. Agrachev, D. Barilari and U. Boscain, On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. Partial Differential Equations, 43 (2012), 355-388. |
[2] |
L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann., 318 (2000), 527-555. |
[3] |
L. Ambrosio and P. Tilli, "Topics on Analysis in Metric Spaces," volume 25 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2004. |
[4] |
P. Assouad, Plongements lipschitziens dans $R^n$, Bull. Soc. Math. France, 111 (1983), 429-448. |
[5] |
A. Bellaïche, The tangent space in sub-Riemannian geometry, In "Sub-Riemannian Geometry," volume 144 of Progr. Math., pages 1-78. Birkhäuser, Basel, 1996. |
[6] |
A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Ann., 98 (1928), 422-464. |
[7] |
U. Boscain, G. Charlot, R. Ghezzi and M. Sigalotti, Lipschitz classification of two-dimensional almost-Riemannian distances on compact oriented surfaces, preprint, arXiv:1003.4842, to appear on Journal of Geometric Analysis, published online 23 september 2011.
doi: 10.1007/s12220-011-9262-4. |
[8] |
E. Falbel and F. Jean, Measures of transverse paths in sub-Riemannian geometry, J. Anal. Math., 91 (2003), 231-246. |
[9] |
H. Federer, The $(\varphi,k)$ rectifiable subsets of $n$-space, Trans. Amer. Soc., 62 (1947), 114-192. |
[10] |
H. Federer, "Geometric Measure Theory," Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. |
[11] |
B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann., 321 (2001), 479-531. |
[12] |
J.-P. Gauthier, B. Jakubczyk and V. Zakalyukin, Motion planning and fastly oscillating controls, SIAM J. Control Optim., 48 (2009/10), 3433-3448. |
[13] |
J.-P. Gauthier and V. Zakalyukin, On the one-step-bracket-generating motion planning problem, J. Dyn. Control Syst., 11 (2005), 215-235. |
[14] |
M. Gromov, Carnot-Carathéodory spaces seen from within, In "Sub-Riemannian Geometry," volume 144 of Progr. Math., pages 79-323. Birkhäuser, Basel, 1996. |
[15] |
G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325. |
[16] |
F. Jean, "Paths in Sub-Riemannian Geometry," Springer (A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek Eds.), 2000. |
[17] |
F. Jean, Entropy and complexity of a path in sub-Riemannian geometry, ESAIM Control Optim. Calc. Var., 9 (2003), 485-508 (electronic). |
[18] |
B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc., 121 (1994), 113-123. |
[19] |
V. Magnani, Characteristic points, rectifiability and perimeter measure on stratified groups, J. Eur. Math. Soc., 8 (2006), 585-609. |
[20] |
J. M. Marstrand, The $(\varphi, s)$ regular subsets of $n$-space, Trans. Amer. Math. Soc., 113 (1964), 369-392. |
[21] |
P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1996. |
[22] |
P. Mattila, R. Serapioni and F. Serra Cassano, Characterizations of intrinsic rectifiability in Heisenberg groups, Ann. Sc. Norm. Super. Pisa Cl. Sci., 9 (2010), 687-723. |
[23] |
E. F. Moore, Density ratios and $(\phi,1)$ rectifiability in $n$-space, Trans. Amer. Math. Soc., 69 (1950), 324-334. |
[24] |
D. Preiss, Geometry of measures in $R^n$: distribution, rectifiability, and densities, Ann. of Math., 125 (1987), 537-643. |
[25] |
K. Weierstrass, On continuous functions of a real argument that do not have a well-defined differential quotient, G.A. Edgar, Classics on Fractals. Addison-Wesley Publishing Company, 1993. |
show all references
References:
[1] |
A. Agrachev, D. Barilari and U. Boscain, On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. Partial Differential Equations, 43 (2012), 355-388. |
[2] |
L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann., 318 (2000), 527-555. |
[3] |
L. Ambrosio and P. Tilli, "Topics on Analysis in Metric Spaces," volume 25 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2004. |
[4] |
P. Assouad, Plongements lipschitziens dans $R^n$, Bull. Soc. Math. France, 111 (1983), 429-448. |
[5] |
A. Bellaïche, The tangent space in sub-Riemannian geometry, In "Sub-Riemannian Geometry," volume 144 of Progr. Math., pages 1-78. Birkhäuser, Basel, 1996. |
[6] |
A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Ann., 98 (1928), 422-464. |
[7] |
U. Boscain, G. Charlot, R. Ghezzi and M. Sigalotti, Lipschitz classification of two-dimensional almost-Riemannian distances on compact oriented surfaces, preprint, arXiv:1003.4842, to appear on Journal of Geometric Analysis, published online 23 september 2011.
doi: 10.1007/s12220-011-9262-4. |
[8] |
E. Falbel and F. Jean, Measures of transverse paths in sub-Riemannian geometry, J. Anal. Math., 91 (2003), 231-246. |
[9] |
H. Federer, The $(\varphi,k)$ rectifiable subsets of $n$-space, Trans. Amer. Soc., 62 (1947), 114-192. |
[10] |
H. Federer, "Geometric Measure Theory," Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. |
[11] |
B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann., 321 (2001), 479-531. |
[12] |
J.-P. Gauthier, B. Jakubczyk and V. Zakalyukin, Motion planning and fastly oscillating controls, SIAM J. Control Optim., 48 (2009/10), 3433-3448. |
[13] |
J.-P. Gauthier and V. Zakalyukin, On the one-step-bracket-generating motion planning problem, J. Dyn. Control Syst., 11 (2005), 215-235. |
[14] |
M. Gromov, Carnot-Carathéodory spaces seen from within, In "Sub-Riemannian Geometry," volume 144 of Progr. Math., pages 79-323. Birkhäuser, Basel, 1996. |
[15] |
G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325. |
[16] |
F. Jean, "Paths in Sub-Riemannian Geometry," Springer (A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek Eds.), 2000. |
[17] |
F. Jean, Entropy and complexity of a path in sub-Riemannian geometry, ESAIM Control Optim. Calc. Var., 9 (2003), 485-508 (electronic). |
[18] |
B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc., 121 (1994), 113-123. |
[19] |
V. Magnani, Characteristic points, rectifiability and perimeter measure on stratified groups, J. Eur. Math. Soc., 8 (2006), 585-609. |
[20] |
J. M. Marstrand, The $(\varphi, s)$ regular subsets of $n$-space, Trans. Amer. Math. Soc., 113 (1964), 369-392. |
[21] |
P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1996. |
[22] |
P. Mattila, R. Serapioni and F. Serra Cassano, Characterizations of intrinsic rectifiability in Heisenberg groups, Ann. Sc. Norm. Super. Pisa Cl. Sci., 9 (2010), 687-723. |
[23] |
E. F. Moore, Density ratios and $(\phi,1)$ rectifiability in $n$-space, Trans. Amer. Math. Soc., 69 (1950), 324-334. |
[24] |
D. Preiss, Geometry of measures in $R^n$: distribution, rectifiability, and densities, Ann. of Math., 125 (1987), 537-643. |
[25] |
K. Weierstrass, On continuous functions of a real argument that do not have a well-defined differential quotient, G.A. Edgar, Classics on Fractals. Addison-Wesley Publishing Company, 1993. |
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