Journal of Integral Equations and Applications

Boundary integral equations for the transmission eigenvalue problem for Maxwell's equations

Fioralba Cakoni, Houssem Haddar, and Shixu Meng

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Abstract

In this paper, we consider the transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that changes sign inside its support. We formulate the transmission eigenvalue problem as an equivalent homogeneous system of the boundary integral equation and, assuming that the contrast is constant near the boundary of the support of the inhomogeneity, we prove that the operator associated with this system is Fredholm of index zero and depends analytically on the wave number. Then we show the existence of wave numbers that are not transmission eigenvalues which by an application of the analytic Fredholm theory implies that the set of transmission eigenvalues is discrete with positive infinity as the only accumulation point.

Article information

Source
J. Integral Equations Applications, Volume 27, Number 3 (2015), 375-406.

Dates
First available in Project Euclid: 17 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1450388941

Digital Object Identifier
doi:10.1216/JIE-2015-27-3-375

Mathematical Reviews number (MathSciNet)
MR3435806

Zentralblatt MATH identifier
1332.35247

Subjects
Primary: 35J25: Boundary value problems for second-order elliptic equations 45A05: Linear integral equations 45C05: Eigenvalue problems [See also 34Lxx, 35Pxx, 45P05, 47A75] 45Q05: Inverse problems 78A25: Electromagnetic theory, general 78A48: Composite media; random media

Keywords
The transmission eigenvalue problem inverse scattering boundary integral equations Maxwell's equations

Citation

Cakoni, Fioralba; Haddar, Houssem; Meng, Shixu. Boundary integral equations for the transmission eigenvalue problem for Maxwell's equations. J. Integral Equations Applications 27 (2015), no. 3, 375--406. doi:10.1216/JIE-2015-27-3-375. https://projecteuclid.org/euclid.jiea/1450388941


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