Boundary Integral Equations for the Transmission Eigenvalue Problem for Maxwell’s Equations

Fioralba Cakoni 1 Houssem Haddar 2 Shixu Meng 1
2 DeFI - Shape reconstruction and identification
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
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 boundary integral equa- tion, 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.
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Fioralba Cakoni, Houssem Haddar, Shixu Meng. Boundary Integral Equations for the Transmission Eigenvalue Problem for Maxwell’s Equations. Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2015, pp.26. ⟨hal-01214257⟩

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