Spectral properties of an acoustic-elastic transmission eigenvalue problem with applications

TL;DR

This paper investigates the spectral properties of a coupled acoustic-elastic transmission eigenvalue problem, proving eigenvalue existence under certain conditions and revealing boundary localization of eigenfunctions, with applications to metamaterials and inverse problems.

Contribution

It introduces the first proof of eigenvalue existence for this problem and demonstrates boundary localization of eigenfunctions, advancing understanding in coupled-physics spectral analysis.

Findings

Existence of acoustic-elastic transmission eigenvalues under mild conditions

Eigenfunctions tend to localize on the boundary of the domain

Implications for metamaterial design and fluid-structure inverse problems

Abstract

We are concerned with a coupled-physics spectral problem arising in the coupled propagation of acoustic and elastic waves, which is referred to as the acoustic-elastic transmission eigenvalue problem. There are two major contributions in this work which are new to the literature. First, under a mild condition on the medium parameters, we prove the existence of an acoustic-elastic transmission eigenvalue. Second, we establish a geometric rigidity result of the transmission eigenfunctions by showing that they tend to localize on the boundary of the underlying domain. Moreover, we also consider interesting implications of the obtained results to the effective construction of metamaterials by using bubbly elastic structures and to the inverse problem associated with the fluid-structure interaction.