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Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation
Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements
University of Tunis El Manar, National Engineering School of Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia |
In this work, we study the stable determination of time-dependent coefficients appearing in the Schrödinger equation from partial Dirichlet-to-Neumann map measured on an arbitrary part of the boundary. Specifically, we establish stability estimates up to the natural gauge for the magnetic potential.
References:
[1] |
I. B. Aïcha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010.
doi: 10.1088/0266-5611/31/12/125010. |
[2] |
I. B. Aïcha, Stability estimate for an inverse problem for the Schrödinger equation in a magnetic field with time-dependent coefficient, Journal of Mathematical Physics, 58 (2017), 071508.
doi: 10.1063/1.4995606. |
[3] |
I. B. Aïcha and Y. Mejri,
Simultaneous determination of the magnetic field and the electric potential in the Schrödinger equation by a finite number of boundary observations, Journal of Inverse and Ill-posed Problems, 26 (2018), 201-209.
doi: 10.1515/jiip-2017-0028. |
[4] |
M. Bellassoued, Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33 (2017), 055009.,
doi: 10.1088/1361-6420/aa5fc5. |
[5] |
M. Bellassoued and M. Choulli,
Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boundary observation, Journal de Mathématiques Pures et Appliquées, 91 (2009), 233-255.
doi: 10.1016/j.matpur.2008.06.002. |
[6] |
M. Bellassoued and M. Choulli,
Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal., 258 (2010), 161-195.
doi: 10.1016/j.jfa.2009.06.010. |
[7] |
M. Bellassoued and D. D. S. Ferreira, Stable determination of coefficients in the dynamical anisotropic Schrödinger equation from the Dirichlet-to-Neumann map, Inverse Problems, 26 (2010), 125010.
doi: 10.1088/0266-5611/26/12/125010. |
[8] |
M. Bellassoued, Y. Kian and E. Soccorsi,
An inverse stability result for non compactly supported potentials by one arbitrary lateral Neumann observation, J. Differential Equations, 260 (2016), 7535-7562.
doi: 10.1016/j.jde.2016.01.033. |
[9] |
M. Bellassoued, Y. Kian and E. Soccorsi,
An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publications of the Research Institute for Mathematical Sciences, 54 (2018), 679-728.
doi: 10.4171/PRIMS/54-4-1. |
[10] |
M. Bellassoued and I. Rassas,
Stability estimate in the determination of a time-dependent coefficient for hyperbolic equation by partial Dirichlet-to-Neumann map, Applicable Analysis, 98 (2019), 2751-2782.
doi: 10.1080/00036811.2018.1471206. |
[11] |
H. Brezis, Analyse Foncionnelle. Théorie et Applications, Collection mathématiques appliquées pour la maitrise, Masson, Paris, 1993. |
[12] |
M. Choulli and Y. Kian,
Stability of the determination of a time-dependent coefficient in parabolic equations, Mathematical Control and Related Fields, 3 (2013), 143-160.
doi: 10.3934/mcrf.2013.3.143. |
[13] |
M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map, Journal de Mathématiques Pures et Appliquées, 114 (2018), 235–261.
doi: 10.1016/j.matpur.2017.12.003. |
[14] |
M. Choulli, Y. Kian and E. Soccorsi,
Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558.
doi: 10.1137/140986268. |
[15] |
F. Chung,
A partial data result for the magnetic Schrödinger inverse problem, Analysis & PDE, 7 (2014), 117-157.
doi: 10.2140/apde.2014.7.117. |
[16] |
M. Cristofol and E. Soccorsi,
Stability estimate in an inverse problem for non-autonomous magnetic Schrödinger equations, Applicable Analysis, 90 (2011), 1499-1520.
doi: 10.1080/00036811.2010.524161. |
[17] |
D. D. S. Ferreira, J. Sjöstrand, C. E. Kenig and G. Uhlmann,
Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys., 271 (2007), 467-488.
doi: 10.1007/s00220-006-0151-9. |
[18] |
G. Eskin,
Inverse boundary value problems and the Aharonov-Bohm effect, Inverse Problems, 19 (2002), 49-62.
doi: 10.1088/0266-5611/19/1/303. |
[19] |
G. Eskin,
Inverse problems for the Schrödinger operators with electromagnetic potentials in domains with obstacles, Inverse Problems, 19 (2003), 985-998.
doi: 10.1088/0266-5611/19/4/313. |
[20] |
G. Eskin,
Inverse boundary value problems in domains with several obstacles, Inverse Problems, 20 (2004), 1497-1516.
doi: 10.1088/0266-5611/20/5/011. |
[21] |
G. Eskin,
Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737-1758.
doi: 10.1080/03605300701382340. |
[22] |
G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 022105.
doi: 10.1063/1.2841329. |
[23] |
G. Eskin and J. Ralston,
Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy, Comm. Math. Phys., 173 (1995), 199-224.
doi: 10.1007/BF02100187. |
[24] |
O. Y. Imanuvilov and M. Yamamoto,
Lipschitz stability in inverse parabolic problems by Carleman estimate, Inverse Problems, 14 (1998), 1229-1249.
doi: 10.1088/0266-5611/14/5/009. |
[25] |
V. Isakov and Z. Sun,
Stability estimates for hyperbolic inverse problems with local boundary data, Inverse problems, 8 (1992), 193-206.
doi: 10.1088/0266-5611/8/2/003. |
[26] |
Y. Kian,
A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data, Journal of Spectral Theory, 8 (2018), 235-269.
doi: 10.4171/JST/195. |
[27] |
Y. Kian,
Stability in the determination of a time-dependent coefficient for wave equations from partial data, J. Math. Anal. Appl., 436 (2016), 408-428.
doi: 10.1016/j.jmaa.2015.12.018. |
[28] |
Y. Kian,
Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046.
doi: 10.1137/16M1076708. |
[29] |
Y. Kian and E. Soccorsi,
Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 627-647.
doi: 10.1137/18M1197308. |
[30] |
Y. Kian and A. Tetlow, Hölder stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation, preprint, arXiv: 1901.09728. |
[31] |
R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015.
doi: 10.1088/0266-5611/29/9/095015. |
[32] |
B. Schweizer, On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma, in Trends in Applications of Mathematics to Mechanics, Springer INdAM Ser., 27, Springer, Cham, 2018, 65–79.
doi: 10.1007/978-3-319-75940-1_4. |
[33] |
P. Stefanov and G. Uhlmann,
Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358.
doi: 10.1006/jfan.1997.3188. |
[34] |
L. Tzou,
Stability estimates for coefficients of magnetic Schrödinger equation from full and partial boundary measurements, Comm. Partial Differential Equations, 3 (2008), 1911-1952.
doi: 10.1080/03605300802402674. |
show all references
References:
[1] |
I. B. Aïcha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010.
doi: 10.1088/0266-5611/31/12/125010. |
[2] |
I. B. Aïcha, Stability estimate for an inverse problem for the Schrödinger equation in a magnetic field with time-dependent coefficient, Journal of Mathematical Physics, 58 (2017), 071508.
doi: 10.1063/1.4995606. |
[3] |
I. B. Aïcha and Y. Mejri,
Simultaneous determination of the magnetic field and the electric potential in the Schrödinger equation by a finite number of boundary observations, Journal of Inverse and Ill-posed Problems, 26 (2018), 201-209.
doi: 10.1515/jiip-2017-0028. |
[4] |
M. Bellassoued, Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33 (2017), 055009.,
doi: 10.1088/1361-6420/aa5fc5. |
[5] |
M. Bellassoued and M. Choulli,
Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boundary observation, Journal de Mathématiques Pures et Appliquées, 91 (2009), 233-255.
doi: 10.1016/j.matpur.2008.06.002. |
[6] |
M. Bellassoued and M. Choulli,
Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal., 258 (2010), 161-195.
doi: 10.1016/j.jfa.2009.06.010. |
[7] |
M. Bellassoued and D. D. S. Ferreira, Stable determination of coefficients in the dynamical anisotropic Schrödinger equation from the Dirichlet-to-Neumann map, Inverse Problems, 26 (2010), 125010.
doi: 10.1088/0266-5611/26/12/125010. |
[8] |
M. Bellassoued, Y. Kian and E. Soccorsi,
An inverse stability result for non compactly supported potentials by one arbitrary lateral Neumann observation, J. Differential Equations, 260 (2016), 7535-7562.
doi: 10.1016/j.jde.2016.01.033. |
[9] |
M. Bellassoued, Y. Kian and E. Soccorsi,
An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publications of the Research Institute for Mathematical Sciences, 54 (2018), 679-728.
doi: 10.4171/PRIMS/54-4-1. |
[10] |
M. Bellassoued and I. Rassas,
Stability estimate in the determination of a time-dependent coefficient for hyperbolic equation by partial Dirichlet-to-Neumann map, Applicable Analysis, 98 (2019), 2751-2782.
doi: 10.1080/00036811.2018.1471206. |
[11] |
H. Brezis, Analyse Foncionnelle. Théorie et Applications, Collection mathématiques appliquées pour la maitrise, Masson, Paris, 1993. |
[12] |
M. Choulli and Y. Kian,
Stability of the determination of a time-dependent coefficient in parabolic equations, Mathematical Control and Related Fields, 3 (2013), 143-160.
doi: 10.3934/mcrf.2013.3.143. |
[13] |
M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map, Journal de Mathématiques Pures et Appliquées, 114 (2018), 235–261.
doi: 10.1016/j.matpur.2017.12.003. |
[14] |
M. Choulli, Y. Kian and E. Soccorsi,
Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558.
doi: 10.1137/140986268. |
[15] |
F. Chung,
A partial data result for the magnetic Schrödinger inverse problem, Analysis & PDE, 7 (2014), 117-157.
doi: 10.2140/apde.2014.7.117. |
[16] |
M. Cristofol and E. Soccorsi,
Stability estimate in an inverse problem for non-autonomous magnetic Schrödinger equations, Applicable Analysis, 90 (2011), 1499-1520.
doi: 10.1080/00036811.2010.524161. |
[17] |
D. D. S. Ferreira, J. Sjöstrand, C. E. Kenig and G. Uhlmann,
Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys., 271 (2007), 467-488.
doi: 10.1007/s00220-006-0151-9. |
[18] |
G. Eskin,
Inverse boundary value problems and the Aharonov-Bohm effect, Inverse Problems, 19 (2002), 49-62.
doi: 10.1088/0266-5611/19/1/303. |
[19] |
G. Eskin,
Inverse problems for the Schrödinger operators with electromagnetic potentials in domains with obstacles, Inverse Problems, 19 (2003), 985-998.
doi: 10.1088/0266-5611/19/4/313. |
[20] |
G. Eskin,
Inverse boundary value problems in domains with several obstacles, Inverse Problems, 20 (2004), 1497-1516.
doi: 10.1088/0266-5611/20/5/011. |
[21] |
G. Eskin,
Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737-1758.
doi: 10.1080/03605300701382340. |
[22] |
G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 022105.
doi: 10.1063/1.2841329. |
[23] |
G. Eskin and J. Ralston,
Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy, Comm. Math. Phys., 173 (1995), 199-224.
doi: 10.1007/BF02100187. |
[24] |
O. Y. Imanuvilov and M. Yamamoto,
Lipschitz stability in inverse parabolic problems by Carleman estimate, Inverse Problems, 14 (1998), 1229-1249.
doi: 10.1088/0266-5611/14/5/009. |
[25] |
V. Isakov and Z. Sun,
Stability estimates for hyperbolic inverse problems with local boundary data, Inverse problems, 8 (1992), 193-206.
doi: 10.1088/0266-5611/8/2/003. |
[26] |
Y. Kian,
A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data, Journal of Spectral Theory, 8 (2018), 235-269.
doi: 10.4171/JST/195. |
[27] |
Y. Kian,
Stability in the determination of a time-dependent coefficient for wave equations from partial data, J. Math. Anal. Appl., 436 (2016), 408-428.
doi: 10.1016/j.jmaa.2015.12.018. |
[28] |
Y. Kian,
Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046.
doi: 10.1137/16M1076708. |
[29] |
Y. Kian and E. Soccorsi,
Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 627-647.
doi: 10.1137/18M1197308. |
[30] |
Y. Kian and A. Tetlow, Hölder stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation, preprint, arXiv: 1901.09728. |
[31] |
R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015.
doi: 10.1088/0266-5611/29/9/095015. |
[32] |
B. Schweizer, On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma, in Trends in Applications of Mathematics to Mechanics, Springer INdAM Ser., 27, Springer, Cham, 2018, 65–79.
doi: 10.1007/978-3-319-75940-1_4. |
[33] |
P. Stefanov and G. Uhlmann,
Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358.
doi: 10.1006/jfan.1997.3188. |
[34] |
L. Tzou,
Stability estimates for coefficients of magnetic Schrödinger equation from full and partial boundary measurements, Comm. Partial Differential Equations, 3 (2008), 1911-1952.
doi: 10.1080/03605300802402674. |
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