New results on transmission eigenvalues

Fioralba Cakoni; Drossos Gintides

doi:10.3934/ipi.2010.4.39

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2010, Volume 4, Issue 1: 39-48. Doi: 10.3934/ipi.2010.4.39

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New results on transmission eigenvalues

Fioralba Cakoni^1, and
Drossos Gintides^2,

1.
Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716-2553
2.
Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens

Received: May 31, 2009

Revised: October 31, 2009

Published: January 2010

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Abstract

We consider the interior transmission eigenvalue problem corresponding to the inverse scattering problem for an isotropic inhomogeneous medium. We first prove that transmission eigenvalues exist for media with index of refraction greater or less than one without assuming that the contrast is sufficiently large. Then we show that for an arbitrary Lipshitz domain with constant index of refraction there exists an infinite discrete set of transmission eigenvalues that accumulate at infinity. Finally, for the general case of non constant index of refraction we provide a lower and an upper bound for the first transmission eigenvalue in terms of the first transmission eigenvalue for appropriate balls with constant index of refraction.

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Mathematics Subject Classification: Primary: 35R30, 35Q60, 35J40, 78A25.

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