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May  2009, 3(2): 155-172. doi: 10.3934/ipi.2009.3.155

On the existence of transmission eigenvalues

Andreas Kirsch 1,

1. 

University of Karlsruhe, Department of Mathematics, 76128 Karlsruhe

Received  June 2008 Revised  February 2009 Published  May 2009

The investigation of the far field operator and the Factorization Method in inverse scattering theory leads naturally to the study of corresponding interior transmission eigenvalue problems. In contrast to the classical Dirichlet- or Neumann eigenvalue problem for $-\Delta$ in bounded domains these interior transmiision eigenvalue problem fail to be selfadjoint. In general, existence of eigenvalues is an open problem. In this paper we prove existence of eigenvalues for the scalar Helmholtz equation (isotropic and anisotropic cases) and Maxwell's equations under the condition that the contrast of the scattering medium is large enough.
Keywords: Maxwell's equations, Helmholtz equation, eigenvalue problem..
Mathematics Subject Classification: Primary: 35P15, 35J05; Secondary: 35P25, 35Q6.
Citation: Andreas Kirsch. On the existence of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (2) : 155-172. doi: 10.3934/ipi.2009.3.155
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