American Institute of Mathematical Sciences

Advanced
  • 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

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

[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

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

[1]

Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems & 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 & 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 & 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 & 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 & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307
[6]

Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems & Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026
[7]

Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems & Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939
[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 & Continuous Dynamical Systems, 2004, 11 (2&3) : 649-666. doi: 10.3934/dcds.2004.11.649
[10]

A. Doubov, Enrique Fernández-Cara, Manuel González-Burgos, J. H. Ortega. A geometric inverse problem for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1213-1238. doi: 10.3934/dcdsb.2006.6.1213
[11]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159
[12]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014
[13]

Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems & Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021
[14]

Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091
[15]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[16]

Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems & Imaging, 2019, 13 (5) : 1023-1044. doi: 10.3934/ipi.2019046
[17]

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 & Imaging, , () : -. doi: 10.3934/ipi.2021049
[18]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006
[19]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225
[20]

Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4207-4222. doi: 10.3934/dcdsb.2018133

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences