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We prove the existence of an infinite discrete set of transmission eigenvalues corresponding to the scattering problem for isotropic and anisotropic inhomogeneous media for both the Helmholtz and Maxwell's equations. Our discussion includes the case of the interior transmission problem for an inhomogeneous medium with cavities, i.e. subregions with contrast zero.

We survey the linear sampling method for solving the inverse scattering problem for time-harmonic electromagnetic waves at fixed frequency. We consider scattering by an obstacle as well as scattering by an inhomogeneous medium both in R 2 and R 3 . Included in our discussion is the use of regularization methods for ill-posed problems and numerical examples in both two and three dimensions.

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We propose and analyze a new formulation of the Linear Sampling Method that uses an exact characterization of the targets shape in terms of the so-called farfield operator (at a fixed frequency). This characterization is based on constructing nearby solutions of the farfield equation using minimizing sequences of a least squares cost functional with an appropriate penalty term. We first provide a general framework for the theoretical foundation of the method in the case of noise-free and noisy measurements operator. We then explicit applications for the case of inhomogeneous inclusions and indicate possible straightforward generalizations. We finally validate the method through some numerical tests and compare the performances with classical LSM and the factorization methods.

We derive different classes of generalized impedance boundary conditions for the scattering problem from strongly absorbing obstacles. Compared to existing works, our construction is based on an asymptotic development of the solution with respect to the medium absorption. Error estimates are derived to validate the accuracy of each condition.

Abstract:We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both scalar Helmholtz equation and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester [15] to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that this contrast is large enough.

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