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December  2018, 12(6): 1309-1342. doi: 10.3934/ipi.2018055

Stability estimates for a magnetic Schrödinger operator with partial data

Leyter Potenciano-Machado , and  Alberto Ruiz

Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain

* Corresponding author: Leyter Potenciano-Machado

Received  July 2016 Revised  February 2018 Published  October 2018

Fund Project: The first author is supported by the Project MTM2011-28198 of Ministerio de Economía y Competividad de España.
The second author is supported by the Project 00MTM2014-57769-C3-1-P of Ministerio de Economía y Competividad de España.

In this paper we study local stability estimates for a magnetic Schrödinger operator with partial data on an open bounded set in dimension $ n≥3$. This is the corresponding stability estimates for the identifiability result obtained by Bukhgeim and Uhlmann [2] in the presence of a magnetic field and when the measurements for the Dirichlet-Neumann map are taken on a neighborhood of the illuminated region of the boundary for functions supported on a neighborhood of the shadow region. We obtain log log-estimates for the magnetic fields and log log log-estimates for the electric potentials.

Keywords: Inverse problems, magnetic Schrödinger operator, Dirichlet-Neumann map, complex geometric optic solutions, Carleman estimates, Radon transform.
Mathematics Subject Classification: Primary: 35R30; Secondary: 42B37.
Citation: Leyter Potenciano-Machado, Alberto Ruiz. Stability estimates for a magnetic Schrödinger operator with partial data. Inverse Problems and Imaging, 2018, 12 (6) : 1309-1342. doi: 10.3934/ipi.2018055
References:
[1]

A. Bernal and J. Cerdà, Complex interpolation of quasi-Banach spaces with an A-convex containing space, Ark. Mat., 29 (1991), 183-201.  doi: 10.1007/BF02384336.

[2]

A. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668.  doi: 10.1081/PDE-120002868.

[3]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Radon transform with restricted data and applications, Adv. in Mathematics, 267 (2014), 523-564.  doi: 10.1016/j.aim.2014.08.009.

[4]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Calderón problem with partial data, J. Differential Equations, 260 (2016), 2457-2489.  doi: 10.1016/j.jde.2015.10.007.

[5]

P. Caro and V. Pohjola, Stability estimates for an inverse problem for the magnetic Schrödinger operator, International Math. Research Notices, 21 (2015), 11083-11116.  doi: 10.1093/imrn/rnv020.

[6]

F. J. Chung, A Partial Data Result for the Magnetic Schrödinger Inverse Problem, Analysis and PDE, 7 (2014), 117-157.  doi: 10.2140/apde.2014.7.117.

[7]

D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Mathematics, 178 (2009), 119-171.  doi: 10.1007/s00222-009-0196-4.

[8]

D. Dos Santos Ferreira, C. E. Kenig, J. Sjöstrand and G. Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Mathematical Physics, 271 (2007), 467-488.  doi: 10.1007/s00220-006-0151-9.

[9]

D. Faraco and K. Rogers, The Sobolev norm of characteristic functions with applications to the Calderón Inverse Problem, Quart. J. Math., 64 (2013), 133-147.  doi: 10.1093/qmath/har039.

[10]

H. Heck and J. N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems, 22 (2006), 1787-1796.  doi: 10.1088/0266-5611/22/5/015.

[11]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Annals Math., 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567.

[12]

K. Krupchyk and G. Uhlmann, Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential, Comm. Math. Phys., 327 (2014), 993-1009.  doi: 10.1007/s00220-014-1942-z.

[13]

A. Nachman and B. Street, Reconstruction in the Calderón problem with partial data, Comm. Partial Differential Equations, 35 (2010), 375-390.  doi: 10.1080/03605300903296322.

[14]

G. Nakamura, Z. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388.  doi: 10.1007/BF01460996.

[15]

(CL32) F. Natterer, The Mathematics of Computerized Tomography, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001. doi: 10.1137/1.9780898719284.

[16]

L. Potenciano-Machado, Inverse Bundary Value Problems for the Magnetic Schrödinger Operator, Ph.D thesis, Universidad Autónoma de Madrid, 2017.

[17]

M. Salo, Inverse problems for nonsmooth first-order perturbations of the Laplacian, Academi Scientiarum Fennic. Annales Mathematica Dissertationes, 139 (2004), 67pp.

[18]

W. Sickel, Pointwise multipliers of Lizorkin-Triebel spaces, in The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, (eds. J. Rossmann, P. Takáč and G. Wildenhain), Birkhäuser Basel, 110 (1999), 295–321. doi: 10.1007/978-3-0348-8672-7_17.

[19]

Z. Sun, An inverse boundary value problem for the Schrödinger operator with vector potentials, Trans. Amer. Math. Soc., 338 (1992), 953-969.  doi: 10.2307/2154438.

[20]

L. Tzou, Stability estimates for coefficients of the magnetic Schrödinger equation from full and partial boundary measurements, Comm. Partial Differential Equations, 33 (2008), 1911-1952.  doi: 10.1080/03605300802402674.

show all references

References:
[1]

A. Bernal and J. Cerdà, Complex interpolation of quasi-Banach spaces with an A-convex containing space, Ark. Mat., 29 (1991), 183-201.  doi: 10.1007/BF02384336.

[2]

A. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668.  doi: 10.1081/PDE-120002868.

[3]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Radon transform with restricted data and applications, Adv. in Mathematics, 267 (2014), 523-564.  doi: 10.1016/j.aim.2014.08.009.

[4]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Calderón problem with partial data, J. Differential Equations, 260 (2016), 2457-2489.  doi: 10.1016/j.jde.2015.10.007.

[5]

P. Caro and V. Pohjola, Stability estimates for an inverse problem for the magnetic Schrödinger operator, International Math. Research Notices, 21 (2015), 11083-11116.  doi: 10.1093/imrn/rnv020.

[6]

F. J. Chung, A Partial Data Result for the Magnetic Schrödinger Inverse Problem, Analysis and PDE, 7 (2014), 117-157.  doi: 10.2140/apde.2014.7.117.

[7]

D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Mathematics, 178 (2009), 119-171.  doi: 10.1007/s00222-009-0196-4.

[8]

D. Dos Santos Ferreira, C. E. Kenig, J. Sjöstrand and G. Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Mathematical Physics, 271 (2007), 467-488.  doi: 10.1007/s00220-006-0151-9.

[9]

D. Faraco and K. Rogers, The Sobolev norm of characteristic functions with applications to the Calderón Inverse Problem, Quart. J. Math., 64 (2013), 133-147.  doi: 10.1093/qmath/har039.

[10]

H. Heck and J. N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems, 22 (2006), 1787-1796.  doi: 10.1088/0266-5611/22/5/015.

[11]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Annals Math., 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567.

[12]

K. Krupchyk and G. Uhlmann, Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential, Comm. Math. Phys., 327 (2014), 993-1009.  doi: 10.1007/s00220-014-1942-z.

[13]

A. Nachman and B. Street, Reconstruction in the Calderón problem with partial data, Comm. Partial Differential Equations, 35 (2010), 375-390.  doi: 10.1080/03605300903296322.

[14]

G. Nakamura, Z. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388.  doi: 10.1007/BF01460996.

[15]

(CL32) F. Natterer, The Mathematics of Computerized Tomography, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001. doi: 10.1137/1.9780898719284.

[16]

L. Potenciano-Machado, Inverse Bundary Value Problems for the Magnetic Schrödinger Operator, Ph.D thesis, Universidad Autónoma de Madrid, 2017.

[17]

M. Salo, Inverse problems for nonsmooth first-order perturbations of the Laplacian, Academi Scientiarum Fennic. Annales Mathematica Dissertationes, 139 (2004), 67pp.

[18]

W. Sickel, Pointwise multipliers of Lizorkin-Triebel spaces, in The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, (eds. J. Rossmann, P. Takáč and G. Wildenhain), Birkhäuser Basel, 110 (1999), 295–321. doi: 10.1007/978-3-0348-8672-7_17.

[19]

Z. Sun, An inverse boundary value problem for the Schrödinger operator with vector potentials, Trans. Amer. Math. Soc., 338 (1992), 953-969.  doi: 10.2307/2154438.

[20]

L. Tzou, Stability estimates for coefficients of the magnetic Schrödinger equation from full and partial boundary measurements, Comm. Partial Differential Equations, 33 (2008), 1911-1952.  doi: 10.1080/03605300802402674.

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