The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on possible location of the transmission eigenvalues. If the index of refraction n(x) is real, we get a result on the existence of infinitely many positive ITEs and the Weyl type lower bound on its counting function. All the results are obtained under the assumption that n(x) − 1 does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle. Some results on the discreteness and localization of the spectrum are obtained for complex valued n(x).
The classical ∂-method has been generalized recently [13], [14] to be used in the presence of exceptional points. We apply this generalization to solve Dirac inverse scattering problem with weak assumptions on smoothness of potentials. As a consequence, this provides an effective method of reconstruction of complex-valued one time differentiable conductivities in the inverse impedance tomography problem.
The paper concerns the discreteness of the eigenvalues and the solvability of the interior transmission problem for anisotropic media. Conditions for the ellipticity of the problem are written explicitly, and it is shown that they do not guarantee the discreteness of the eigenvalues. Some simple sufficient conditions for the discreteness and solvability are found. They are expressed in terms of the values of the anisotropy matrix and the refraction index at the boundary of the domain. The discreteness of the eigenvalues and the solvability of the interior transmission problem are shown if a small perturbation is applied to the refraction index.
The paper contains the Weyl formula for the counting function of the interior transmission problem when the latter is parameter-elliptic. Branching billiard trajectories are constructed, and the second term of the Weyl asymptotics is estimated from above under some conditions on the set of periodic billiard trajectories.
The paper contains lower bounds on the counting function of the positive eigenvalues of the interior transmission problem when the latter is elliptic. In particular, these bounds justify the existence of an infinite set of interior transmission eigenvalues and provide asymptotic estimates from above on the counting function for the large values of the wave number. They also lead to certain important upper estimates on the first few interior transmission eigenvalues. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle.
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