Stability estimates for a magnetic Schrödinger operator with partial data

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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 & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	(CL32) F. Natterer, The Mathematics of Computerized Tomography, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001. doi: 10.1137/1.9780898719284. Google Scholar
[16]	L. Potenciano-Machado, Inverse Bundary Value Problems for the Magnetic Schrödinger Operator, Ph.D thesis, Universidad Autónoma de Madrid, 2017.Google Scholar
[17]	M. Salo, Inverse problems for nonsmooth first-order perturbations of the Laplacian, Academi Scientiarum Fennic. Annales Mathematica Dissertationes, 139 (2004), 67pp. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	(CL32) F. Natterer, The Mathematics of Computerized Tomography, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001. doi: 10.1137/1.9780898719284. Google Scholar
[16]	L. Potenciano-Machado, Inverse Bundary Value Problems for the Magnetic Schrödinger Operator, Ph.D thesis, Universidad Autónoma de Madrid, 2017.Google Scholar
[17]	M. Salo, Inverse problems for nonsmooth first-order perturbations of the Laplacian, Academi Scientiarum Fennic. Annales Mathematica Dissertationes, 139 (2004), 67pp. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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