Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem

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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

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

Received March 2014 Revised August 2014 Published November 2014

Abstract
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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. 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, (1980), 65. Google Scholar
[3]	F. J. Chung, A partial data result for the magnetic Schrödinger inverse problem,, Anal. and PDE, 7 (2014), 117. doi: 10.2140/apde.2014.7.117. Google Scholar
[4]	F. J. Chung, Partial data for the Neumann-to-Dirichlet map,, preprint, (2012). Google Scholar
[5]	F. J. Chung, M. Salo and L. Tzou, Partial data inverse problems for the Hodge Laplacian,, preprint, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (2013). Google Scholar
[19]	M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian,, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004). Google Scholar
[20]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153. doi: 10.2307/1971291. Google Scholar
[21]	M. E. Taylor, Partial Differential Equations I: Basic Theory,, Applied Mathematical Sciences, (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). doi: 10.1088/0266-5611/25/12/123011. Google Scholar
[23]	M. Zworski, Semiclassical Analysis,, Graduate Studies in Mathematics, (2012). Google Scholar

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References:

[1]	A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data,, Comm. PDE, 27 (2002), 653. 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, (1980), 65. Google Scholar
[3]	F. J. Chung, A partial data result for the magnetic Schrödinger inverse problem,, Anal. and PDE, 7 (2014), 117. doi: 10.2140/apde.2014.7.117. Google Scholar
[4]	F. J. Chung, Partial data for the Neumann-to-Dirichlet map,, preprint, (2012). Google Scholar
[5]	F. J. Chung, M. Salo and L. Tzou, Partial data inverse problems for the Hodge Laplacian,, preprint, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (2013). Google Scholar
[19]	M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian,, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004). Google Scholar
[20]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153. doi: 10.2307/1971291. Google Scholar
[21]	M. E. Taylor, Partial Differential Equations I: Basic Theory,, Applied Mathematical Sciences, (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). doi: 10.1088/0266-5611/25/12/123011. Google Scholar
[23]	M. Zworski, Semiclassical Analysis,, Graduate Studies in Mathematics, (2012). Google Scholar

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