On a local geometric property of the generalized elastic transmission eigenfunctions and application

TL;DR

This paper investigates the geometric structure of elastic transmission eigenfunctions, proving their local vanishing near corners under certain regularity conditions, and applies these findings to improve inverse elastic scattering problem identifiability.

Contribution

It introduces a generalized elastic transmission eigenvalue problem and establishes local vanishing properties of eigenfunctions near corners, advancing inverse scattering theory.

Findings

Eigenfunctions vanish locally around corners under regularity conditions

New unique identifiability results for inverse elastic problems

Application of geometric properties to inverse scattering with single measurement

Abstract

Consider the nonlinear and completely continuous scattering map \[ \mathcal{S}\big((\Omega; \lambda, \mu, V), \mathbf{u}^i\big)=\mathbf{u}_t^\infty(\hat{\mathbf{x}}), \quad \hat{\mathbf{x}}\in\mathbb{S}^{n-1}, \] which sends an inhomogeneous elastic scatterer $(\Omega; \lambda, \mu, V)$ to its far-field pattern $\mathbf{u}_t^\infty$ due to an incident wave field $\mathbf{u}^i$ via the Lam\'e system. Here, $(\lambda, \mu, V)$ signifies the medium configuration of an elastic scatterer that is compactly supported in $\Omega$. In this paper, we are concerned with the intrinsic geometric structure of the kernel space of $\mathcal{S}$, which is of fundamental importance to the theory of inverse scattering and invisibility cloaking for elastic waves and has received considerable attention recently. It turns out that the study is contained in analysing the geometric properties of a certain non-selfadjoint and non-elliptic transmission eigenvalue problem. We propose a generalized elastic transmission eigenvalue problem and prove that the transmission eigenfunctions vanish locally around a corner of $\partial\Omega$ under generic regularity criteria. The regularity criteria are characerized by the H\"older continuity or a certain Fourier extension property of the transmission eigenfunctions. As an interesting and significant application, we apply the local geometric property to derive several novel unique identifiability results for a longstanding inverse elastic problem by a single far-field measurement.