On the transmission eigenvalues for scattering by a clamped planar region

TL;DR

This paper introduces a new transmission eigenvalue problem for scattering by a clamped cavity in a thin elastic material, proving eigenvalue recovery from far field data and demonstrating results through numerical experiments.

Contribution

It presents a novel transmission eigenvalue problem in all of ^2 for elastic scattering, with theoretical proofs and numerical validation.

Findings

Eigenvalues can be recovered from far field data.

Transmission eigenvalues are discrete.

Numerical experiments support theoretical results.

Abstract

In this paper, we consider a new transmission eigenvalue problem derived from the scattering by a clamped cavity in a thin elastic material. Scattering in a thin elastic material can be modeled by the Kirchhoff--Love infinite plate problem. This results in a biharmonic scattering problem that can be handled by operator splitting. The main novelty of this transmission eigenvalue problem is that it is posed in all of $\mathbb{R}^2$. This adds analytical and computational difficulties in studying this eigenvalue problem. Here, we prove that the eigenvalues can be recovered from the far field data as well as discreteness of the transmission eigenvalues. We provide some numerical experiments via boundary integral equations to demonstrate the theoretical results. We also conjecture monotonicity with respect to the measure of the scatterer from our numerical experiments.