Geometrical characterizations of radiating and non-radiating elastic sources and mediums with applications

TL;DR

This paper explores the geometric properties of elastic wave sources and scatterers, establishing conditions for radiation, uniqueness of support determination from measurements, and properties of elastic eigenfunctions, with applications in inverse scattering.

Contribution

It introduces novel geometric characterizations of elastic sources and scatterers, including conditions for radiation and uniqueness results, using advanced analytical techniques.

Findings

Scatterers with small support or high-curvature points radiate at all frequencies.

Unique determination of scatterer support from a single far-field measurement.

New insights into the geometric properties of elastic transmission eigenfunctions.

Abstract

In this paper, we investigate two types of time-harmonic elastic wave scattering problems. The first one involves the scattered wave generated by an active elastic source with compact support. The second one concerns elastic wave scattering caused by an inhomogeneous medium, also with compact support. We derive several novel quantitative results concerning the geometrical properties of the underlying scatterer, the associated source or incident wave field, and the physical parameters. In particular, we show that a scatterer with either a small support or high-curvature boundary points must radiate at any frequency. These qualitative characterizations allow us to establish several local and global uniqueness results for determining the support of the source or medium scatterer from a single far-field measurement. Furthermore, we reveal new geometric properties of elastic transmission eigenfunctions. To derive a quantitative relationship between the intensity of a radiating or non-radiating source and the diameter of its support, we utilize the Helmholtz decomposition, the translation-invariant $L^2$-norm estimate for the Lam\'e operator, and global energy estimates. Another pivotal technical approach combines complex geometric optics (CGO) solutions with local regularity estimates, facilitating microlocal analysis near admissible $K$-curvature boundary points.