American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation
  • IPI Home
  • This Issue
  • Next Article
    Stability of boundary distance representation and reconstruction of Riemannian manifolds
February  2007, 1(1): 159-179. doi: 10.3934/ipi.2007.1.159

An integral equation approach and the interior transmission problem for Maxwell's equations

Andreas Kirsch 1,

1. 

University of Karlsruhe, Department of Mathematics, 76128 Karlsruhe, Germany

Received  September 2006 Published  January 2007

In the first part of this paper we recall the direct scattering problem for time harmonic electromagnetic fields where arbitrary incident fields are scattered by a medium described by a space dependent permittivity, permeability, and conductivity. We present an integral equation approach and recall its basic features. In the second part we investigate the corresponding interior transmission eigenvalue problem and prove that the spectrum is discrete. Finally, we study the inhomogeneous interior transmission problem and show that it is uniquely solvable provided $k^2$ is not an interior eigenvalue.
Keywords: scattering theory, integral equation, Maxwell's equations, eigenvalue problem..
Mathematics Subject Classification: Primary: 78A46, 35P30; Secondary: 35Q60, 45E0.
Citation: 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
[1]

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
[2]

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
[3]

W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431
[4]

Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257
[5]

Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455
[6]

Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435
[7]

Changxing Miao, Jiqiang Zheng. Scattering theory for energy-supercritical Klein-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2073-2094. doi: 10.3934/dcdss.2016085
[8]

Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261
[9]

Gang Bao, Bin Hu, Peijun Li, Jue Wang. Analysis of time-domain Maxwell's equations in biperiodic structures. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 259-286. doi: 10.3934/dcdsb.2019181
[10]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473
[11]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185
[12]

Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials. Electronic Research Archive, 2020, 28 (2) : 961-976. doi: 10.3934/era.2020051
[13]

B. L. G. Jonsson. Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations. Inverse Problems & Imaging, 2009, 3 (3) : 405-452. doi: 10.3934/ipi.2009.3.405
[14]

Cleverson R. da Luz, Gustavo Alberto Perla Menzala. Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 547-558. doi: 10.3934/dcdss.2009.2.547
[15]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557
[16]

Jeremy L. Marzuola. Dispersive estimates using scattering theory for matrix Hamiltonian equations. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 995-1035. doi: 10.3934/dcds.2011.30.995
[17]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607
[18]

Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229
[19]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903
[20]

Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (110)
  • HTML views (0)
  • Cited by (33)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences