# Existence of transmission eigenvalues for biharmonic scattering by a clamped planar region

## TL;DR

This paper investigates a new transmission eigenvalue problem arising from the scattering of elastic plates, proving the existence of infinitely many real eigenvalues and exploring their relation to classical Laplacian eigenvalues.

## Contribution

It introduces and analyzes a novel clamped transmission eigenvalue problem for biharmonic operators in elastic scattering, establishing the existence of infinitely many eigenvalues.

## Key findings

- Existence of infinitely many real transmission eigenvalues.
- Relationship between transmission eigenvalues and Laplacian eigenvalues.
- Development of a variational approach for the problem.

## Abstract

In this paper, we study the so-called clamped transmission eigenvalue problem. This is a new transmission eigenvalue problem that is derived from the scattering of an impenetrable clamped obstacle in a thin elastic plate. The scattering problem is modeled by a biharmonic wave operator given by the Kirchhoff--Love infinite plate problem in the frequency domain. These scattering problems have not been studied to the extent of other models. Unlike other transmission eigenvalue problems, the problem studied here is a system of homogeneous PDEs defined in all of $\mathbb{R}^2$. This provides unique analytical and computational difficulties when studying the clamped transmission eigenvalue problem. We are able to prove that there exist infinitely many real clamped transmission eigenvalues. This is done by studying the equivalent variational formulation. We also investigate the relationship of the clamped transmission eigenvalues to the Dirichlet and Neumann eigenvalues of the negative Laplacian for the bounded scattering obstacle.