-
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
Stability estimates for a magnetic Schrödinger operator with partial data
Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain |
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 [
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. |
[1] |
Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials. Discrete and 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 and 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 and Continuous Dynamical Systems, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264 |
[4] |
Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731 |
[5] |
Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013 |
[6] |
Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete and 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 and 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 and 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 and Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139 |
[10] |
Mourad Bellassoued, Zouhour Rezig. Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1061-1084. doi: 10.3934/dcdss.2021158 |
[11] |
Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems and Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745 |
[12] |
Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems and Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034 |
[13] |
Xinchi Huang, Masahiro Yamamoto. Carleman estimates for a magnetohydrodynamics system and application to inverse source problems. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022005 |
[14] |
Gan Lu, Weiming Liu. Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials. Communications on Pure and Applied Analysis, 2017, 16 (6) : 1957-1975. doi: 10.3934/cpaa.2017096 |
[15] |
Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889 |
[16] |
Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933 |
[17] |
Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems and Imaging, 2018, 12 (1) : 1-28. doi: 10.3934/ipi.2018001 |
[18] |
Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109 |
[19] |
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. |
[20] |
Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems and Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721 |
2021 Impact Factor: 1.483
Tools
Metrics
Other articles
by authors
[Back to Top]