American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Simultaneous reconstruction and segmentation with the Mumford-Shah functional for electron tomography
  • IPI Home
  • This Issue
  • Next Article
    Reconstruction of the coefficients of a star graph from observations of its vertices
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

[1]

Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 661-671. doi: 10.3934/dcdss.2017033
[2]

Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959
[3]

Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264
[4]

Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013
[5]

Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731
[6]

Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631
[7]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169
[8]

Ru-Yu Lai. Global uniqueness for an inverse problem for the magnetic Schrödinger operator. Inverse Problems & Imaging, 2011, 5 (1) : 59-73. doi: 10.3934/ipi.2011.5.59
[9]

Victor Isakov, Jenn-Nan Wang. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139
[10]

Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745
[11]

Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems & Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034
[12]

Gan Lu, Weiming Liu. Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1957-1975. doi: 10.3934/cpaa.2017096
[13]

Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889
[14]

Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933
[15]

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
[16]

Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109
[17]

Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79.

[18]

Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721
[19]

Xinchi Huang, Atsushi Kawamoto. Inverse problems for a half-order time-fractional diffusion equation in arbitrary dimension by Carleman estimates. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021040
[20]

Peng Gao. Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications. Evolution Equations & Control Theory, 2018, 7 (3) : 465-499. doi: 10.3934/eect.2018023

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (163)
  • HTML views (214)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences