American Institute of Mathematical Sciences

Advanced
November  2014, 8(4): 959-989. doi: 10.3934/ipi.2014.8.959

Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem

Francis J. Chung 1,

1. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States

Received  March 2014 Revised  August 2014 Published  November 2014

We show that an electric potential and magnetic field can be uniquely determined by partial boundary measurements of the Neumann-to-Dirichlet map of the associated magnetic Schrödinger operator. This improves upon the results in [4] by including the determination of a magnetic field. The main technical advance is an improvement on the Carleman estimate in [4]. This allows the construction of complex geometrical optics solutions with greater regularity, which are needed to deal with the first order term in the operator. This improved regularity of CGO solutions may have applications in the study of inverse problems in systems of equations with partial boundary data.
Keywords: Carleman estimate., magnetic Schrödinger equation, inverse problems, Neumann-Dirichlet map, Calderón problem, partial data.
Mathematics Subject Classification: Primary: 35R3.
Citation: 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
References:
[1]

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

[2]

A. P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemática, Río de Janeiro, 1980, 65-73.  Google Scholar

[3]

F. J. Chung, A partial data result for the magnetic Schrödinger inverse problem, Anal. and PDE, 7 (2014), 117-157. doi: 10.2140/apde.2014.7.117.  Google Scholar

[4]

F. J. Chung, Partial data for the Neumann-to-Dirichlet map, preprint, arXiv:1211.0211, (2012). Google Scholar

[5]

F. J. Chung, M. Salo and L. Tzou, Partial data inverse problems for the Hodge Laplacian, preprint, arXiv:1310.4616, (2013). Google Scholar

[6]

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

[7]

D. Dos Santos Ferreira, C. E. Kenig, J. Sjöstrand 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.  Google Scholar

[8]

N. Hyvönen, P. Piiroinen and O. Seiskari, Point measurements for a Neumann-to-Dirichlet map and the Calderón problem in the plane, SIAM J. Math. Anal., 44 (2012), 3526-3536. arXiv:1204.0346. doi: 10.1137/120872164.  Google Scholar

[9]

O. Imanuvilov, G. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc., 23 (2010), 655-691. doi: 10.1090/S0894-0347-10-00656-9.  Google Scholar

[10]

O. Imanuvilov, G. Uhlmann and M. Yamamoto, Inverse boundary value problem by partial data for Neumann-to-Dirichlet-map in two dimensions,, preprint, ().   Google Scholar

[11]

V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105. doi: 10.3934/ipi.2007.1.95.  Google Scholar

[12]

C. E. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Anal. and PDE, 6 (2013), 2003-2048. doi: 10.2140/apde.2013.6.2003.  Google Scholar

[13]

C. E. Kenig and M. Salo, Recent progress in the Calderón problem with partial data, Contemp. Math., 615 (2014), 193-222. doi: 10.1090/conm/615/12245.  Google Scholar

[14]

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

[15]

K. Krupchyk, M. Lassas and G. Uhlmann, Inverse problems with partial data for a magnetic Schrödinger operator in an infinite slab and on a bounded domain, Comm. Math. Phys., 312 (2012), 87-126. doi: 10.1007/s00220-012-1431-1.  Google Scholar

[16]

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

[17]

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

[18]

V. Pohjola, An inverse problem for the magnetic Schrödinger operator on a half space with partial data, preprint, arXiv:1302.7265, (2013). Google Scholar

[19]

M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004), 67 pp.  Google Scholar

[20]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

[21]

M. E. Taylor, Partial Differential Equations I: Basic Theory, Applied Mathematical Sciences, 115, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[22]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inv. Prob., 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

[23]

M. Zworski, Semiclassical Analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, Providence, RI, 2012.  Google Scholar

show all references

References:
[1]

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

[2]

A. P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemática, Río de Janeiro, 1980, 65-73.  Google Scholar

[3]

F. J. Chung, A partial data result for the magnetic Schrödinger inverse problem, Anal. and PDE, 7 (2014), 117-157. doi: 10.2140/apde.2014.7.117.  Google Scholar

[4]

F. J. Chung, Partial data for the Neumann-to-Dirichlet map, preprint, arXiv:1211.0211, (2012). Google Scholar

[5]

F. J. Chung, M. Salo and L. Tzou, Partial data inverse problems for the Hodge Laplacian, preprint, arXiv:1310.4616, (2013). Google Scholar

[6]

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

[7]

D. Dos Santos Ferreira, C. E. Kenig, J. Sjöstrand 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.  Google Scholar

[8]

N. Hyvönen, P. Piiroinen and O. Seiskari, Point measurements for a Neumann-to-Dirichlet map and the Calderón problem in the plane, SIAM J. Math. Anal., 44 (2012), 3526-3536. arXiv:1204.0346. doi: 10.1137/120872164.  Google Scholar

[9]

O. Imanuvilov, G. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc., 23 (2010), 655-691. doi: 10.1090/S0894-0347-10-00656-9.  Google Scholar

[10]

O. Imanuvilov, G. Uhlmann and M. Yamamoto, Inverse boundary value problem by partial data for Neumann-to-Dirichlet-map in two dimensions,, preprint, ().   Google Scholar

[11]

V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105. doi: 10.3934/ipi.2007.1.95.  Google Scholar

[12]

C. E. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Anal. and PDE, 6 (2013), 2003-2048. doi: 10.2140/apde.2013.6.2003.  Google Scholar

[13]

C. E. Kenig and M. Salo, Recent progress in the Calderón problem with partial data, Contemp. Math., 615 (2014), 193-222. doi: 10.1090/conm/615/12245.  Google Scholar

[14]

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

[15]

K. Krupchyk, M. Lassas and G. Uhlmann, Inverse problems with partial data for a magnetic Schrödinger operator in an infinite slab and on a bounded domain, Comm. Math. Phys., 312 (2012), 87-126. doi: 10.1007/s00220-012-1431-1.  Google Scholar

[16]

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

[17]

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

[18]

V. Pohjola, An inverse problem for the magnetic Schrödinger operator on a half space with partial data, preprint, arXiv:1302.7265, (2013). Google Scholar

[19]

M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004), 67 pp.  Google Scholar

[20]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

[21]

M. E. Taylor, Partial Differential Equations I: Basic Theory, Applied Mathematical Sciences, 115, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[22]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inv. Prob., 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

[23]

M. Zworski, Semiclassical Analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, Providence, RI, 2012.  Google Scholar

[1]

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
[2]

Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469
[3]

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
[4]

María Ángeles García-Ferrero, Angkana Rüland, Wiktoria Zatoń. Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021049
[5]

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
[6]

Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems & Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021
[7]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[8]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307
[9]

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
[10]

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
[11]

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
[12]

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
[13]

Yanni Guo, Genqi Xu, Yansha Guo. Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2491-2507. doi: 10.3934/dcdsb.2016057
[14]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[15]

Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems & Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449
[16]

Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems & Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063
[17]

Matteo Santacesaria. Note on Calderón's inverse problem for measurable conductivities. Inverse Problems & Imaging, 2019, 13 (1) : 149-157. doi: 10.3934/ipi.2019008
[18]

Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems & Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49
[19]

Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221
[20]

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

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences