Absence of the analytic continuation of elastic transmission eigenfunctions at rectangular corners

TL;DR

This paper investigates elastic wave scattering by objects with rectangular corners, showing that such scatterers always produce nontrivial scattering despite having interior transmission eigenvalues, and introduces new analytical techniques for inverse problems.

Contribution

It demonstrates the non-existence of analytic continuation of elastic transmission eigenfunctions at rectangular corners and offers a novel approach for inverse elastic medium problems with polygonal shapes.

Findings

Rectangular corners cause nontrivial elastic scattering.

Interior transmission eigenvalues do not imply trivial far field operator.

New decomposition method for elastic Lamé operator enhances inverse problem analysis.

Abstract

We study time harmonic scattering problems in linear elasticity in $\mathbb{R}^{2}$. We show that certain penetrable scatterers with rectangular corners scatter every incident wave nontrivially. Even though these scatterers have interior transmission eigenvalues, the far field operator has a trivial kernel at every real frequency. Our approach relies on a special decomposition of the elastic Lam\'e operator and also provides an alternative idea for treating inverse elastic medium problems with a general polygonal support.