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    Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation
October  2020, 14(5): 841-865. doi: 10.3934/ipi.2020039

Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements

Mourad Bellassoued , and  Oumaima Ben Fraj

University of Tunis El Manar, National Engineering School of Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia

* Corresponding author: Mourad Bellassoued

Received  December 2019 Revised  March 2020 Published  July 2020

Fund Project: The authors are supported by the Tunisian Research Program PHC-UTIQUE 19G-1507

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.

Keywords: Stability estimate, Schrödinger equation, partial data, Dirichlet-to-Neumann map, FBI transformation.
Mathematics Subject Classification: Primary: 35R30, 35Q60; Secondary: 35L53.
Citation: Mourad Bellassoued, Oumaima Ben Fraj. Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements. Inverse Problems and Imaging, 2020, 14 (5) : 841-865. doi: 10.3934/ipi.2020039
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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