Abstract

We consider the interior transmission problem with one complex-valued refraction index, that is, with a damping term which does not vanish on the boundary. Under the condition that all geodesics reach the boundary, for a class of strictly concave domains, we obtain a transmission eigenvalue-free region of the form $\{\lambda \in ℂ:C_N{(|\lambda |+1)}^{-N}\le |\mathrm{Im}<!--FUNCTION\; APPLICATION-->\lambda |\le C{(|\lambda |+1)}^{-1}\}$, where $N>1$ is arbitrary. Under extra conditions on the coefficients we get a larger transmission eigenvalue-free region of the form $\{\lambda \in ℂ:C_N{(|\lambda |+1)}^{-N}\le |\mathrm{Im}<!--FUNCTION\; APPLICATION-->\lambda |\le C\}$.

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