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November  2014, 8(4): 1117-1137. doi: 10.3934/ipi.2014.8.1117

Calderón problem for Maxwell's equations in cylindrical domain

Oleg Yu. Imanuvilov 1, and  Masahiro Yamamoto 2,

1. 

Department of Mathematics, Colorado State University,101 Weber Building, Fort Colins, CO 80523-1784, United States
2. 

Department of Mathematical Sciences, The University of Tokyo, Komaba Meguro Tokyo 153-8914

Received  December 2013 Revised  September 2014 Published  November 2014

We prove some uniqueness results in determination of the conductivity, the permeability and the permittivity of Maxwell's equations in a cylindrical domain $\Omega \times (0,L)$ from partial boundary map. More specifically, for an arbitrarily given subboundary $\Gamma_0 \subset \partial\Omega$, we prove that the coefficients of Maxwell's equations can be uniquely determined in the subdomain $(\Omega \setminus$ [the convex hull of $\Gamma_0])$ $ \times (0,L)$ by the boundary map only for inputs vanishing on $\Gamma_0 \times (0,L)$.
Keywords: complex geometric optics solutions, Calderón problem, Carleman estimate, Maxwell's equations., inverse problem.
Mathematics Subject Classification: Primary: 35R30, 35F35; Secondary: 35Q6.
Citation: Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117
References:
[1]

J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, Duke Math. J., 55 (1987), 943-948. doi: 10.1215/S0012-7094-87-05547-5.  Google Scholar

[2]

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

[3]

P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data, Comm. P.D.E., 34 (2009), 1425-1464. doi: 10.1080/03605300903296272.  Google Scholar

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R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1990.  Google Scholar

[5]

S. Helgason, Integral Geometry and Radon Transforms, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-1-4419-6055-9.  Google Scholar

[6]

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

[7]

O. Imanuvilov and M. Yamamoto, Inverse boundary value problem for the Schrödinger equation in cylindrical domain by partial boundary data, Inverse Problems , 29 (2013), 045002, 8pp. doi: 10.1088/0266-5611/29/4/045002.  Google Scholar

[8]

O. Imanuvilov and M. Yamamoto, Uniqueness for inverse boundary value problems by Dirichlet-to-Neumann map on subboundaries, Milan J. Math., 81 (2013), 187-258. doi: 10.1007/s00032-013-0205-3.  Google Scholar

[9]

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

[10]

C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Mathematical Subject Classification, 6 (2013), 2003-2048. arXiv:1211.1054. doi: 10.2140/apde.2013.6.2003.  Google Scholar

[11]

A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar

[12]

P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke . Math. J., 70 (1993), 617-653. doi: 10.1215/S0012-7094-93-07014-7.  Google Scholar

[13]

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

show all references

References:
[1]

J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, Duke Math. J., 55 (1987), 943-948. doi: 10.1215/S0012-7094-87-05547-5.  Google Scholar

[2]

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

[3]

P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data, Comm. P.D.E., 34 (2009), 1425-1464. doi: 10.1080/03605300903296272.  Google Scholar

[4]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1990.  Google Scholar

[5]

S. Helgason, Integral Geometry and Radon Transforms, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-1-4419-6055-9.  Google Scholar

[6]

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

[7]

O. Imanuvilov and M. Yamamoto, Inverse boundary value problem for the Schrödinger equation in cylindrical domain by partial boundary data, Inverse Problems , 29 (2013), 045002, 8pp. doi: 10.1088/0266-5611/29/4/045002.  Google Scholar

[8]

O. Imanuvilov and M. Yamamoto, Uniqueness for inverse boundary value problems by Dirichlet-to-Neumann map on subboundaries, Milan J. Math., 81 (2013), 187-258. doi: 10.1007/s00032-013-0205-3.  Google Scholar

[9]

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

[10]

C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Mathematical Subject Classification, 6 (2013), 2003-2048. arXiv:1211.1054. doi: 10.2140/apde.2013.6.2003.  Google Scholar

[11]

A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar

[12]

P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke . Math. J., 70 (1993), 617-653. doi: 10.1215/S0012-7094-93-07014-7.  Google Scholar

[13]

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

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