The interior transmission eigenvalue problem for elastic waves in media with obstacles

Fioralba Cakoni; Pu-Zhao Kow; Jenn-Nan Wang

doi:10.3934/ipi.2020075

Article Contents

2021, Volume 15, Issue 3: 445-474. Doi: 10.3934/ipi.2020075

This issue Previous Article Some application examples of minimization based formulations of inverse problems and their regularization Next Article Tensor train rank minimization with nonlocal self-similarity for tensor completion

The interior transmission eigenvalue problem for elastic waves in media with obstacles

1.
Department of Mathematics, Rutgers University, New Brunswick, USA
2.
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
3.
Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan

^* Corresponding author: Pu-Zhao Kow
^* Corresponding author: Pu-Zhao Kow

Received: June 2020

Revised: September 2020

Early access: November 2020

Published: June 2021

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Abstract

In this paper, we investigate the interior transmission eigenvalue problem for elastic waves propagating outside a sound-soft or a sound-hard obstacle surrounded by an anisotropic layer. This study is motivated by the inverse problem of identifying an object embedded in an inhomogeneous media in the presence of elastic waves. Our analysis of this non-selfadjoint eigenvalue problem relies on the weak formulation of involved boundary value problems and some fundamental tools in functional analysis.

Keywords:

Interior transmission eigenvalue problem,
elastic waves,
anisotropic inhomogeneous media,
media with obstacles,
non-selfadjoint eigenvalue problem.

Mathematics Subject Classification: Primary: 35P99, 35Q74; Secondary: 47A53, 47A75.

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Figure 1. Plot of $ F(k) $ in (21) with $ \mu = 1 $, $ \lambda = 1 $ and $ n = 1/2 $ (GNU Octave)

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Figure 2. Plot of $ F(k) $ in (22) with $ \mu = 1 $, $ \lambda = 1 $ and $ n = 1/2 $ (GNU Octave)

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