Calderón problem for Maxwell's equations in cylindrical domain

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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

Abstract
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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. doi: 10.1215/S0012-7094-87-05547-5.
[2]	A. P. Calderón, On an inverse boundary value problem,, in Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980), 65.
[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. doi: 10.1080/03605300903296272.
[4]	R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology,, Springer-Verlag, (1990).
[5]	S. Helgason, Integral Geometry and Radon Transforms,, Springer-Verlag, (2011). doi: 10.1007/978-1-4419-6055-9.
[6]	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.
[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). doi: 10.1088/0266-5611/29/4/045002.
[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. doi: 10.1007/s00032-013-0205-3.
[9]	V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Problems and Imaging, 1 (2007), 95. doi: 10.3934/ipi.2007.1.95.
[10]	C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications,, Mathematical Subject Classification, 6 (2013), 2003. doi: 10.2140/apde.2013.6.2003.
[11]	A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math., 143 (1996), 71. doi: 10.2307/2118653.
[12]	P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics,, Duke . Math. J., 70 (1993), 617. doi: 10.1215/S0012-7094-93-07014-7.
[13]	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.

show all references

References:

[1]	J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms,, Duke Math. J., 55 (1987), 943. doi: 10.1215/S0012-7094-87-05547-5.
[2]	A. P. Calderón, On an inverse boundary value problem,, in Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980), 65.
[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. doi: 10.1080/03605300903296272.
[4]	R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology,, Springer-Verlag, (1990).
[5]	S. Helgason, Integral Geometry and Radon Transforms,, Springer-Verlag, (2011). doi: 10.1007/978-1-4419-6055-9.
[6]	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.
[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). doi: 10.1088/0266-5611/29/4/045002.
[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. doi: 10.1007/s00032-013-0205-3.
[9]	V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Problems and Imaging, 1 (2007), 95. doi: 10.3934/ipi.2007.1.95.
[10]	C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications,, Mathematical Subject Classification, 6 (2013), 2003. doi: 10.2140/apde.2013.6.2003.
[11]	A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math., 143 (1996), 71. doi: 10.2307/2118653.
[12]	P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics,, Duke . Math. J., 70 (1993), 617. doi: 10.1215/S0012-7094-93-07014-7.
[13]	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.

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