# American Institute of Mathematical Sciences

November  2015, 9(4): 935-950. doi: 10.3934/ipi.2015.9.935

## 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

Received  February 2015 Revised  June 2015 Published  October 2015

In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential $\alpha$ and a potential $U$. For brevity, this type of systems are called $\mathcal{MP}$-systems. On simple $\mathcal{MP}$-systems, we consider both the boundary rigidity problem and scattering rigidity problem. Unlike the cases of geodesic or magnetic systems, knowing boundary action functions or scattering relations for only one energy level is insufficient to uniquely determine a simple $\mathcal{MP}$-system up to natural obstructions, even under the assumption that the boundary restriction of the system is given, and we provide some counterexamples. By reducing an $\mathcal{MP}$-system to the corresponding magnetic system and applying the results of [6] on simple magnetic systems, we prove rigidity results for metrics in a given conformal class, for simple real analytic $\mathcal{MP}$-systems and for simple two-dimensional $\mathcal{MP}$-systems.
Citation: Yernat M. Assylbekov, Hanming Zhou. Boundary and scattering rigidity problems in the presence of a magnetic field and a potential. Inverse Problems & Imaging, 2015, 9 (4) : 935-950. doi: 10.3934/ipi.2015.9.935
##### References:
 [1] D. V. Anosov and Y. G. Sinai, Certain smooth ergodic systems [Russian],, Uspekhi Mat. Nauk, 22 (1967), 107. Google Scholar [2] V. I. Arnold, Some remarks on flows of line elements and frames,, Sov. Math. Dokl., 138 (1961), 255. Google Scholar [3] V. I. Arnold and A. B. Givental, Symplectic Geometry,, Dynamical Systems IV, (1990). Google Scholar [4] C. B. Croke, Rigidity and distance between boundary points,, J. Diff. Geom., 33 (1991), 445. Google Scholar [5] C. B. Croke, Rigidity theorems in Riemannian geometry,, in Geometric Methods in Inverse Problems and PDE Control, (2004), 47. doi: 10.1007/978-1-4684-9375-7_4. Google Scholar [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. doi: 10.1016/j.aim.2007.05.014. Google Scholar [7] N. S. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation,, Inverse Problems and Imaging, 4 (2010), 397. doi: 10.3934/ipi.2010.4.397. Google Scholar [8] M. L. Gerver and N. S. Nadirashvili, Inverse problem of mechanics at high energies,, (Russian) Comput. Seismology, 15 (1983), 118. Google Scholar [9] P. Herreros, Scattering boundary rigidity in the presence of a magnetic field,, Comm. Anal. Geom., 20 (2012), 501. doi: 10.4310/CAG.2012.v20.n3.a3. Google Scholar [10] P. Herreros and J. Vargo, Scattering rigidity for analytic Riemannian manifolds with a possible magnetic field,, J. Geom. Anal., 21 (2011), 641. doi: 10.1007/s12220-010-9162-z. Google Scholar [11] A. Jollivet, On inverse problems in electromagnetic field in classical mechanics at fixed energy,, J. Geom. Anal., 17 (2007), 275. doi: 10.1007/BF02930725. Google Scholar [12] V. V. Kozlov, Calculus of variations in the large and classical mechanics,, (Russian) Uspekhi Mat. Nauk, 40 (1985), 33. Google Scholar [13] R. Michel, Sur la rigidité imposée par la longueur des géodésiques,, Invent. Math., 65 (1981), 71. doi: 10.1007/BF01389295. Google Scholar [14] R. G. Mukhometov, On a problem of reconstructing Riemannian metrics,, (Russian) Sibirsk. Mat. Zh., 22 (1981), 119. Google Scholar [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. Google Scholar [16] S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II,, (Russian) Funktsional. Anal. i Prilozhen., 15 (1981), 37. Google Scholar [17] S. P. Novikov, Hamiltonian formalism and a multivalued analogue of Morse theory,, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3. Google Scholar [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. Google Scholar [19] R. G. Novikov, Small angle scattering and X-ray transform in classical mechanics,, Ark. Mat., 37 (1999), 141. doi: 10.1007/BF02384831. Google Scholar [20] G. P. Paternain and M. Paternain, Anosov geodesic flows and twisted symplectic structures,, in International Conference on Dynamical Systems (Montevideo, (1995), 132. Google Scholar [21] L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid,, Ann. of Math., 161 (2005), 1093. doi: 10.4007/annals.2005.161.1093. Google Scholar [22] V. A. Sharafutdinov, Integral Geometry of Tensor Fields,, VSP, (1994). doi: 10.1515/9783110900095. Google Scholar [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. doi: 10.1215/S0012-7094-04-12332-2. Google Scholar [24] P. Stefanov and G.Uhlmann, Boundary rigidity and stability for generic simple metrics,, J. Amer. Math. Soc., 18 (2005), 975. doi: 10.1090/S0894-0347-05-00494-7. Google Scholar [25] P. Stefanov and G. Uhlmann, Recent progress on the boundary rigidity problem,, Electron. Res. Announc. Amer. Math. Soc., 11 (2005), 64. doi: 10.1090/S1079-6762-05-00148-4. Google Scholar [26] P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data,, preprint, (). doi: 10.1090/jams/846. Google Scholar

show all references

##### References:
 [1] D. V. Anosov and Y. G. Sinai, Certain smooth ergodic systems [Russian],, Uspekhi Mat. Nauk, 22 (1967), 107. Google Scholar [2] V. I. Arnold, Some remarks on flows of line elements and frames,, Sov. Math. Dokl., 138 (1961), 255. Google Scholar [3] V. I. Arnold and A. B. Givental, Symplectic Geometry,, Dynamical Systems IV, (1990). Google Scholar [4] C. B. Croke, Rigidity and distance between boundary points,, J. Diff. Geom., 33 (1991), 445. Google Scholar [5] C. B. Croke, Rigidity theorems in Riemannian geometry,, in Geometric Methods in Inverse Problems and PDE Control, (2004), 47. doi: 10.1007/978-1-4684-9375-7_4. Google Scholar [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. doi: 10.1016/j.aim.2007.05.014. Google Scholar [7] N. S. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation,, Inverse Problems and Imaging, 4 (2010), 397. doi: 10.3934/ipi.2010.4.397. Google Scholar [8] M. L. Gerver and N. S. Nadirashvili, Inverse problem of mechanics at high energies,, (Russian) Comput. Seismology, 15 (1983), 118. Google Scholar [9] P. Herreros, Scattering boundary rigidity in the presence of a magnetic field,, Comm. Anal. Geom., 20 (2012), 501. doi: 10.4310/CAG.2012.v20.n3.a3. Google Scholar [10] P. Herreros and J. Vargo, Scattering rigidity for analytic Riemannian manifolds with a possible magnetic field,, J. Geom. Anal., 21 (2011), 641. doi: 10.1007/s12220-010-9162-z. Google Scholar [11] A. Jollivet, On inverse problems in electromagnetic field in classical mechanics at fixed energy,, J. Geom. Anal., 17 (2007), 275. doi: 10.1007/BF02930725. Google Scholar [12] V. V. Kozlov, Calculus of variations in the large and classical mechanics,, (Russian) Uspekhi Mat. Nauk, 40 (1985), 33. Google Scholar [13] R. Michel, Sur la rigidité imposée par la longueur des géodésiques,, Invent. Math., 65 (1981), 71. doi: 10.1007/BF01389295. Google Scholar [14] R. G. Mukhometov, On a problem of reconstructing Riemannian metrics,, (Russian) Sibirsk. Mat. Zh., 22 (1981), 119. Google Scholar [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. Google Scholar [16] S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II,, (Russian) Funktsional. Anal. i Prilozhen., 15 (1981), 37. Google Scholar [17] S. P. Novikov, Hamiltonian formalism and a multivalued analogue of Morse theory,, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3. Google Scholar [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. Google Scholar [19] R. G. Novikov, Small angle scattering and X-ray transform in classical mechanics,, Ark. Mat., 37 (1999), 141. doi: 10.1007/BF02384831. Google Scholar [20] G. P. Paternain and M. Paternain, Anosov geodesic flows and twisted symplectic structures,, in International Conference on Dynamical Systems (Montevideo, (1995), 132. Google Scholar [21] L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid,, Ann. of Math., 161 (2005), 1093. doi: 10.4007/annals.2005.161.1093. Google Scholar [22] V. A. Sharafutdinov, Integral Geometry of Tensor Fields,, VSP, (1994). doi: 10.1515/9783110900095. Google Scholar [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. doi: 10.1215/S0012-7094-04-12332-2. Google Scholar [24] P. Stefanov and G.Uhlmann, Boundary rigidity and stability for generic simple metrics,, J. Amer. Math. Soc., 18 (2005), 975. doi: 10.1090/S0894-0347-05-00494-7. Google Scholar [25] P. Stefanov and G. Uhlmann, Recent progress on the boundary rigidity problem,, Electron. Res. Announc. Amer. Math. Soc., 11 (2005), 64. doi: 10.1090/S1079-6762-05-00148-4. Google Scholar [26] P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data,, preprint, (). doi: 10.1090/jams/846. Google Scholar
 [1] Hanming Zhou. Lens rigidity with partial data in the presence of a magnetic field. Inverse Problems & 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 & Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002 [3] Plamen Stefanov and Gunther Uhlmann. Recent progress on the boundary rigidity problem. Electronic Research Announcements, 2005, 11: 64-70. [4] Gilles Carbou, Stéphane Labbé, Emmanuel Trélat. Smooth control of nanowires by means of a magnetic field. Communications on Pure & Applied Analysis, 2009, 8 (3) : 871-879. doi: 10.3934/cpaa.2009.8.871 [5] Nurlan Dairbekov, Gunther Uhlmann. Reconstructing the metric and magnetic field from the scattering relation. Inverse Problems & Imaging, 2010, 4 (3) : 397-409. doi: 10.3934/ipi.2010.4.397 [6] 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 [7] Vladimir Georgiev, Atanas Stefanov, Mirko Tarulli. Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 771-786. doi: 10.3934/dcds.2007.17.771 [8] Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109 [9] 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 [10] Jiying Ma, Dongmei Xiao. Nonlinear dynamics of a mathematical model on action potential duration and calcium transient in paced cardiac cells. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2377-2396. doi: 10.3934/dcdsb.2013.18.2377 [11] Mingqi Xiang, Patrizia Pucci, Marco Squassina, Binlin Zhang. Nonlocal Schrödinger-Kirchhoff equations with external magnetic field. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1631-1649. doi: 10.3934/dcds.2017067 [12] 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 [13] Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a phase field system with a possibly singular potential. Mathematical Control & Related Fields, 2016, 6 (1) : 95-112. doi: 10.3934/mcrf.2016.6.95 [14] Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a conserved phase field system with a possibly singular potential. Evolution Equations & Control Theory, 2018, 7 (1) : 95-116. doi: 10.3934/eect.2018006 [15] 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 & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737 [16] Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems & Imaging, 2018, 12 (1) : 1-28. doi: 10.3934/ipi.2018001 [17] Alex H. Ardila. Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field. Communications on Pure & Applied Analysis, 2018, 17 (1) : 163-175. doi: 10.3934/cpaa.2018010 [18] Martin Seehafer. A local existence result for a plasma physics model containing a fully coupled magnetic field. Kinetic & Related Models, 2009, 2 (3) : 503-520. doi: 10.3934/krm.2009.2.503 [19] Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644 [20] Jianjun Chen, Wancheng Sheng. The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2018, 17 (1) : 127-142. doi: 10.3934/cpaa.2018008

2018 Impact Factor: 1.469