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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 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.

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