Calderón problem for Maxwell's equations in cylindrical domain

Previous Article
Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map
IPI Home
This Issue
Next Article
An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method

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
Related Papers

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 and 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.
[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.
[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.
[4]	R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1990.
[5]	S. Helgason, Integral Geometry and Radon Transforms, Springer-Verlag, Berlin, 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), 655-691. 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), 045002, 8pp. 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), 187-258. 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), 95-105. 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), 2003-2048. arXiv:1211.1054. 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), 71-96. doi: 10.2307/2118653.
[12]	P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke . Math. J., 70 (1993), 617-653. 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), 153-169. 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), 943-948. 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, 65-73, Soc. Brasil. Mat., Río de Janeiro, 1980.
[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.
[4]	R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1990.
[5]	S. Helgason, Integral Geometry and Radon Transforms, Springer-Verlag, Berlin, 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), 655-691. 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), 045002, 8pp. 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), 187-258. 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), 95-105. 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), 2003-2048. arXiv:1211.1054. 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), 71-96. doi: 10.2307/2118653.
[12]	P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke . Math. J., 70 (1993), 617-653. 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), 153-169. doi: 10.2307/1971291.

[1]	Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems and Imaging, 2014, 8 (4) : 991-1012. doi: 10.3934/ipi.2014.8.991
[2]	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
[3]	Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems and Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49
[4]	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
[5]	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
[6]	Henrik Garde, Nuutti Hyvönen. Reconstruction of singular and degenerate inclusions in Calderón's problem. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022021
[7]	Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems and Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026
[8]	Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001
[9]	Daomin Cao, Ezzat S. Noussair, Shusen Yan. On the profile of solutions for an elliptic problem arising in nonlinear optics. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 649-666. doi: 10.3934/dcds.2004.11.649
[10]	Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems and Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939
[11]	A. Doubov, Enrique Fernández-Cara, Manuel González-Burgos, J. H. Ortega. A geometric inverse problem for the Boussinesq system. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1213-1238. doi: 10.3934/dcdsb.2006.6.1213
[12]	Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control and Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014
[13]	Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems and Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159
[14]	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
[15]	Kim Knudsen, Aksel Kaastrup Rasmussen. Direct regularized reconstruction for the three-dimensional Calderón problem. Inverse Problems and Imaging, 2022, 16 (4) : 871-894. doi: 10.3934/ipi.2022002
[16]	Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091
[17]	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
[18]	Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems and Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[19]	Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems and Imaging, 2019, 13 (5) : 1023-1044. doi: 10.3934/ipi.2019046
[20]	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

2020 Impact Factor: 1.639

Tools

Metrics

Other articles
by authors

on AIMS
Oleg Yu. Imanuvilov Masahiro Yamamoto
on Google Scholar
Oleg Yu. Imanuvilov Masahiro Yamamoto

[Back to Top]