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

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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

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

Received September 2006 Published January 2007

Abstract
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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

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