Analysis of the transmission eigenvalue problem for biharmonic scattering considering penetrable scatterers

TL;DR

The paper addresses transmission eigenvalues for biharmonic scattering by penetrable obstacles in a two-dimensional Kirchhoff–Love plate model, where some energy transmits into the medium. It develops a variational formulation and operator-theoretic framework to study the TE spectrum, proving discreteness and the existence of infinitely many real eigenvalues, and establishing monotonicity of the first eigenvalue with respect to the refractive index $n$. Numerical validation is provided via separation of variables and boundary integral equations, confirming the theoretical results across various geometries and index values. The findings extend the classical acoustic TE theory to biharmonic elasticity, offering a rigorous foundation for inverse problems and potential reconstruction methods in elasticity and plate theories.

Abstract

In this paper, we provide an analytical study of the transmission eigenvalue problem in the context of biharmonic scattering with a penetrable obstacle. We will assume that the underlying physical model is given by an infinite elastic two--dimensional Kirchhoff--Love plate in $\mathbb{R}^2$, where the plate's thickness is small relative to the wavelength of the incident wave. In previous studies, transmission eigenvalues have been studied for acoustic scattering, whereas in this case, we consider biharmonic scattering. We prove the existence and discreteness of the transmission eigenvalues as well as study the dependence on the refractive index. We are able to prove the monotonicity of the first transmission eigenvalue with respect to the refractive index. Lastly, we provide numerical experiments to validate the theoretical work.

Theorems & Definitions (21)

• Theorem 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1
• proof
• Theorem 4.1
• Theorem 4.2