-
Previous Article
The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media
- IPI Home
- This Issue
- Next Article
Boundary and scattering rigidity problems in the presence of a magnetic field and a potential
1. | Department of Mathematics, University of Washington, Seattle, WA 98195-4350 |
2. | DPMMS, Centre for Mathematical Sciences, Cambridge CB3 0WB, United Kingdom |
References:
[1] |
D. V. Anosov and Y. G. Sinai, Certain smooth ergodic systems [Russian], Uspekhi Mat. Nauk, 22 (1967), 107-172. |
[2] |
V. I. Arnold, Some remarks on flows of line elements and frames, Sov. Math. Dokl., 138 (1961), 255-257. |
[3] |
V. I. Arnold and A. B. Givental, Symplectic Geometry, Dynamical Systems IV, Encyclopaedia of Mathematical Sciences, Springer Verlag, Berlin, 1990. |
[4] |
C. B. Croke, Rigidity and distance between boundary points, J. Diff. Geom., 33 (1991), 445-464. |
[5] |
C. B. Croke, Rigidity theorems in Riemannian geometry, in Geometric Methods in Inverse Problems and PDE Control, IMA Vol. Math. Appl., 137, Springer, New York, 2004, 47-72.
doi: 10.1007/978-1-4684-9375-7_4. |
[6] |
N. S. Dairbekov, G. P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609.
doi: 10.1016/j.aim.2007.05.014. |
[7] |
N. S. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation, Inverse Problems and Imaging, 4 (2010), 397-409.
doi: 10.3934/ipi.2010.4.397. |
[8] |
M. L. Gerver and N. S. Nadirashvili, Inverse problem of mechanics at high energies, (Russian) Comput. Seismology, 15 (1983), 118-125. |
[9] |
P. Herreros, Scattering boundary rigidity in the presence of a magnetic field, Comm. Anal. Geom., 20 (2012), 501-528.
doi: 10.4310/CAG.2012.v20.n3.a3. |
[10] |
P. Herreros and J. Vargo, Scattering rigidity for analytic Riemannian manifolds with a possible magnetic field, J. Geom. Anal., 21 (2011), 641-664.
doi: 10.1007/s12220-010-9162-z. |
[11] |
A. Jollivet, On inverse problems in electromagnetic field in classical mechanics at fixed energy, J. Geom. Anal., 17 (2007), 275-319.
doi: 10.1007/BF02930725. |
[12] |
V. V. Kozlov, Calculus of variations in the large and classical mechanics, (Russian) Uspekhi Mat. Nauk, 40 (1985), 33-60, 237. |
[13] |
R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., 65 (1981), 71-83.
doi: 10.1007/BF01389295. |
[14] |
R. G. Mukhometov, On a problem of reconstructing Riemannian metrics, (Russian) Sibirsk. Mat. Zh., 22 (1981), 119-135, 237. |
[15] |
R. G. Mukhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in $n$-dimensional space, (Russian) Dokl. Akad. Nauk SSSR, 243 (1978), 41-44. |
[16] |
S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II, (Russian) Funktsional. Anal. i Prilozhen., 15 (1981), 37-52, 96. |
[17] |
S. P. Novikov, Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3-49, 248. |
[18] |
S. P. Novikov and I. Shmel'tser, Periodic solutions of the Kirchhoff equations for the free motion of a rigid body in a liquid, and the extended Lyusternik-Schnirelmann-Morse theory. I, (Russian) Funktsional. Anal. i Prilozhen., 15 (1981), 54-66. |
[19] |
R. G. Novikov, Small angle scattering and X-ray transform in classical mechanics, Ark. Mat., 37 (1999), 141-169.
doi: 10.1007/BF02384831. |
[20] |
G. P. Paternain and M. Paternain, Anosov geodesic flows and twisted symplectic structures, in International Conference on Dynamical Systems (Montevideo, 1995) (eds. F. Ledrappier, J. Lewowicz and S. Newhouse), Pitman Research Notes in Math., 362, Longman, Harlow, 1996, 132-145. |
[21] |
L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math., 161 (2005), 1093-1110.
doi: 10.4007/annals.2005.161.1093. |
[22] |
V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, 1994.
doi: 10.1515/9783110900095. |
[23] |
P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467.
doi: 10.1215/S0012-7094-04-12332-2. |
[24] |
P. Stefanov and G.Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003.
doi: 10.1090/S0894-0347-05-00494-7. |
[25] |
P. Stefanov and G. Uhlmann, Recent progress on the boundary rigidity problem, Electron. Res. Announc. Amer. Math. Soc., 11 (2005), 64-70.
doi: 10.1090/S1079-6762-05-00148-4. |
[26] |
P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data,, preprint, ().
doi: 10.1090/jams/846. |
show all references
References:
[1] |
D. V. Anosov and Y. G. Sinai, Certain smooth ergodic systems [Russian], Uspekhi Mat. Nauk, 22 (1967), 107-172. |
[2] |
V. I. Arnold, Some remarks on flows of line elements and frames, Sov. Math. Dokl., 138 (1961), 255-257. |
[3] |
V. I. Arnold and A. B. Givental, Symplectic Geometry, Dynamical Systems IV, Encyclopaedia of Mathematical Sciences, Springer Verlag, Berlin, 1990. |
[4] |
C. B. Croke, Rigidity and distance between boundary points, J. Diff. Geom., 33 (1991), 445-464. |
[5] |
C. B. Croke, Rigidity theorems in Riemannian geometry, in Geometric Methods in Inverse Problems and PDE Control, IMA Vol. Math. Appl., 137, Springer, New York, 2004, 47-72.
doi: 10.1007/978-1-4684-9375-7_4. |
[6] |
N. S. Dairbekov, G. P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609.
doi: 10.1016/j.aim.2007.05.014. |
[7] |
N. S. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation, Inverse Problems and Imaging, 4 (2010), 397-409.
doi: 10.3934/ipi.2010.4.397. |
[8] |
M. L. Gerver and N. S. Nadirashvili, Inverse problem of mechanics at high energies, (Russian) Comput. Seismology, 15 (1983), 118-125. |
[9] |
P. Herreros, Scattering boundary rigidity in the presence of a magnetic field, Comm. Anal. Geom., 20 (2012), 501-528.
doi: 10.4310/CAG.2012.v20.n3.a3. |
[10] |
P. Herreros and J. Vargo, Scattering rigidity for analytic Riemannian manifolds with a possible magnetic field, J. Geom. Anal., 21 (2011), 641-664.
doi: 10.1007/s12220-010-9162-z. |
[11] |
A. Jollivet, On inverse problems in electromagnetic field in classical mechanics at fixed energy, J. Geom. Anal., 17 (2007), 275-319.
doi: 10.1007/BF02930725. |
[12] |
V. V. Kozlov, Calculus of variations in the large and classical mechanics, (Russian) Uspekhi Mat. Nauk, 40 (1985), 33-60, 237. |
[13] |
R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., 65 (1981), 71-83.
doi: 10.1007/BF01389295. |
[14] |
R. G. Mukhometov, On a problem of reconstructing Riemannian metrics, (Russian) Sibirsk. Mat. Zh., 22 (1981), 119-135, 237. |
[15] |
R. G. Mukhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in $n$-dimensional space, (Russian) Dokl. Akad. Nauk SSSR, 243 (1978), 41-44. |
[16] |
S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II, (Russian) Funktsional. Anal. i Prilozhen., 15 (1981), 37-52, 96. |
[17] |
S. P. Novikov, Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3-49, 248. |
[18] |
S. P. Novikov and I. Shmel'tser, Periodic solutions of the Kirchhoff equations for the free motion of a rigid body in a liquid, and the extended Lyusternik-Schnirelmann-Morse theory. I, (Russian) Funktsional. Anal. i Prilozhen., 15 (1981), 54-66. |
[19] |
R. G. Novikov, Small angle scattering and X-ray transform in classical mechanics, Ark. Mat., 37 (1999), 141-169.
doi: 10.1007/BF02384831. |
[20] |
G. P. Paternain and M. Paternain, Anosov geodesic flows and twisted symplectic structures, in International Conference on Dynamical Systems (Montevideo, 1995) (eds. F. Ledrappier, J. Lewowicz and S. Newhouse), Pitman Research Notes in Math., 362, Longman, Harlow, 1996, 132-145. |
[21] |
L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math., 161 (2005), 1093-1110.
doi: 10.4007/annals.2005.161.1093. |
[22] |
V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, 1994.
doi: 10.1515/9783110900095. |
[23] |
P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467.
doi: 10.1215/S0012-7094-04-12332-2. |
[24] |
P. Stefanov and G.Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003.
doi: 10.1090/S0894-0347-05-00494-7. |
[25] |
P. Stefanov and G. Uhlmann, Recent progress on the boundary rigidity problem, Electron. Res. Announc. Amer. Math. Soc., 11 (2005), 64-70.
doi: 10.1090/S1079-6762-05-00148-4. |
[26] |
P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data,, preprint, ().
doi: 10.1090/jams/846. |
[1] |
Hanming Zhou. Lens rigidity with partial data in the presence of a magnetic field. Inverse Problems and Imaging, 2018, 12 (6) : 1365-1387. doi: 10.3934/ipi.2018057 |
[2] |
Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations and Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002 |
[3] |
Qihong Shi, Congming Peng, Qingxuan Wang. Blowup results for the fractional Schrödinger equation without gauge invariance. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021304 |
[4] |
Plamen Stefanov and Gunther Uhlmann. Recent progress on the boundary rigidity problem. Electronic Research Announcements, 2005, 11: 64-70. |
[5] |
Gilles Carbou, Stéphane Labbé, Emmanuel Trélat. Smooth control of nanowires by means of a magnetic field. Communications on Pure and Applied Analysis, 2009, 8 (3) : 871-879. doi: 10.3934/cpaa.2009.8.871 |
[6] |
Nurlan Dairbekov, Gunther Uhlmann. Reconstructing the metric and magnetic field from the scattering relation. Inverse Problems and Imaging, 2010, 4 (3) : 397-409. doi: 10.3934/ipi.2010.4.397 |
[7] |
Uri Bader, Roman Muchnik. Boundary unitary representations-irreducibility and rigidity. Journal of Modern Dynamics, 2011, 5 (1) : 49-69. doi: 10.3934/jmd.2011.5.49 |
[8] |
J. I. Díaz, J. F. Padial. On a free-boundary problem modeling the action of a limiter on a plasma. Conference Publications, 2007, 2007 (Special) : 313-322. doi: 10.3934/proc.2007.2007.313 |
[9] |
Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127 |
[10] |
Vladimir Georgiev, Atanas Stefanov, Mirko Tarulli. Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 771-786. doi: 10.3934/dcds.2007.17.771 |
[11] |
Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109 |
[12] |
Jiying Ma, Dongmei Xiao. Nonlinear dynamics of a mathematical model on action potential duration and calcium transient in paced cardiac cells. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2377-2396. doi: 10.3934/dcdsb.2013.18.2377 |
[13] |
Mingqi Xiang, Patrizia Pucci, Marco Squassina, Binlin Zhang. Nonlocal Schrödinger-Kirchhoff equations with external magnetic field. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1631-1649. doi: 10.3934/dcds.2017067 |
[14] |
Amer Rasheed, Aziz Belmiloudi, Fabrice Mahé. Dynamics of dendrite growth in a binary alloy with magnetic field effect. Conference Publications, 2011, 2011 (Special) : 1224-1233. doi: 10.3934/proc.2011.2011.1224 |
[15] |
Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic and Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011 |
[16] |
Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a phase field system with a possibly singular potential. Mathematical Control and Related Fields, 2016, 6 (1) : 95-112. doi: 10.3934/mcrf.2016.6.95 |
[17] |
Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a conserved phase field system with a possibly singular potential. Evolution Equations and Control Theory, 2018, 7 (1) : 95-116. doi: 10.3934/eect.2018006 |
[18] |
Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics and Games, 2021, 8 (4) : 467-486. doi: 10.3934/jdg.2021014 |
[19] |
Tigran Bakaryan, Rita Ferreira, Diogo Gomes. A potential approach for planning mean-field games in one dimension. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2147-2187. doi: 10.3934/cpaa.2022054 |
[20] |
Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]