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Transmission eigenvalue-free regions near the real axis in the anisotropic case

Georgi Vodev

Abstract

We consider the anisotropic interior transmission problem with one complex-valued refraction index. Under the condition that all geodesics reach the boundary in a finite time, we obtain large regions near the real axis free of transmission eigenvalues.

Transmission eigenvalue-free regions near the real axis in the anisotropic case

Abstract

We consider the anisotropic interior transmission problem with one complex-valued refraction index. Under the condition that all geodesics reach the boundary in a finite time, we obtain large regions near the real axis free of transmission eigenvalues.
Paper Structure (6 sections, 20 theorems, 198 equations)

This paper contains 6 sections, 20 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. A priori estimates under the condition (\ref{['eq:1.6']})
  3. A priori estimates when $m\equiv 0$
  4. The Dirichlet-to-Neumann map
  5. Parametrix of the Dirichlet-to-Neumann map in the elliptic region
  6. Proof of Theorem 1.1

Key Result

Theorem 1.1

Suppose that the conditions (eq:1.3), (eq:1.6) and (eq:1.7) are satisfied. Suppose also that $\Gamma$ is $g_2-$strictly concave. Then for every $N>1$ there exist constants $C, C_N>0$, $C$ being independent of $N$, such that there are no transmission eigenvalues in the region If in addition we assume the condition then there are no transmission eigenvalues in the region

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Theorem 3.1
  • ...and 10 more