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

Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem

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.
Citation: 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
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.

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[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.

[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.

[10]

O. Imanuvilov, G. Uhlmann and M. Yamamoto, Inverse boundary value problem by partial data for Neumann-to-Dirichlet-map in two dimensions, preprint, arXiv:1210.1255.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[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.

[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.

[22]

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

[23]

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

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.

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[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.

[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.

[10]

O. Imanuvilov, G. Uhlmann and M. Yamamoto, Inverse boundary value problem by partial data for Neumann-to-Dirichlet-map in two dimensions, preprint, arXiv:1210.1255.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[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.

[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.

[22]

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

[23]

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

[1]

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

[2]

Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems and 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 and Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034

[4]

Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021060

[5]

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 and Imaging, 2022, 16 (1) : 251-281. doi: 10.3934/ipi.2021049

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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 and Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[17]

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

[18]

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

[19]

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

[20]

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

2021 Impact Factor: 1.483

Metrics

  • PDF downloads (113)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]