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On a coupled-physics transmission eigenvalue problem and its spectral properties with applications

Huaian Diao, Hongyu Liu, Qingle Meng, Li Wang

Abstract

In this paper, we investigate a transmission eigenvalue problem that couples the principles of acoustics and elasticity. This problem naturally arises when studying fluid-solid interactions and constructing bubbly-elastic structures to create metamaterials. We uncover intriguing local geometric structures of the transmission eigenfunctions near the corners of the domains, under typical regularity conditions. As applications, we present novel unique identifiability and visibility results for an inverse problem associated with an acoustoelastic system, which hold practical significance.

On a coupled-physics transmission eigenvalue problem and its spectral properties with applications

Abstract

In this paper, we investigate a transmission eigenvalue problem that couples the principles of acoustics and elasticity. This problem naturally arises when studying fluid-solid interactions and constructing bubbly-elastic structures to create metamaterials. We uncover intriguing local geometric structures of the transmission eigenfunctions near the corners of the domains, under typical regularity conditions. As applications, we present novel unique identifiability and visibility results for an inverse problem associated with an acoustoelastic system, which hold practical significance.

Paper Structure

This paper contains 9 sections, 24 theorems, 162 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical formulation
  3. Motivations and background
  4. Discussion on existing results and summary of main findings
  5. Local geometrical properties near corners of acoustic-elastic transmission eigenfunctions: two-dimensional case
  6. Local geometrical properties near corners of acoustic-elastic transmission eigenfunctions: three dimensional case
  7. Auxiliary lemmas of proving Theorem \ref{['thm:3Ru123']}
  8. The proof of Theorem \ref{['thm:3Ru123']}
  9. Visbility and unique recovery results for the inverse scattering problem

Key Result

Lemma 2.1

Let $a \in \mathbb{C}$ and $\Re(a)>0$. For any given positive numbers $\alpha$ and $h$ satisfying $0<h < e$, if $\Re(a) \geq \frac{2 \alpha}{ e}$, one has as $\Re(a)\rightarrow +\infty.$

Figures (2)

  • Figure 1: Schematic illustration of a 2D corner.
  • Figure 2: Schematic illustration of a 3D edge corner.

Theorems & Definitions (52)

  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 42 more