Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Isaac Harris, Andreas Kleefeld, Heejin Lee

TL;DR

This work establishes the existence of infinitely many real clamped transmission eigenvalues for biharmonic scattering in a planar Kirchhoff–Love plate. By formulating a truncated-domain variational problem with operators $\mathbb{A}_k$ and $\mathbb{B}$ and leveraging a result of Cakoni–TE, the authors prove coercivity of $\mathbb{A}_0$ and the required spectral properties to guarantee discrete real eigenvalues, while linking the first eigenvalue to the Dirichlet spectrum via $k_1^2 \le \lambda_1$. Numerically, the study uses boundary-integral methods to compute clamped, Dirichlet, and Neumann eigenvalues for disk and several nontrivial shapes, revealing interlacing patterns and monotonicity of the first clamped eigenvalue with respect to the obstacle measure. The results enhance understanding of transmission phenomena in biharmonic scattering and suggest directions for extending to other boundary conditions and inverse problems. The methods and findings have potential applications in nondestructive testing and plate scattering analysis.

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.

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

TL;DR

This work establishes the existence of infinitely many real clamped transmission eigenvalues for biharmonic scattering in a planar Kirchhoff–Love plate. By formulating a truncated-domain variational problem with operators and and leveraging a result of Cakoni–TE, the authors prove coercivity of and the required spectral properties to guarantee discrete real eigenvalues, while linking the first eigenvalue to the Dirichlet spectrum via . Numerically, the study uses boundary-integral methods to compute clamped, Dirichlet, and Neumann eigenvalues for disk and several nontrivial shapes, revealing interlacing patterns and monotonicity of the first clamped eigenvalue with respect to the obstacle measure. The results enhance understanding of transmission phenomena in biharmonic scattering and suggest directions for extending to other boundary conditions and inverse problems. The methods and findings have potential applications in nondestructive testing and plate scattering analysis.

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 . 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.

Paper Structure

This paper contains 5 sections, 9 theorems, 77 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Formulation of Transmission Eigenvalue Problem
  3. Existence of the Transmission Eigenvalues
  4. Numerical Investigation
  5. Conclusion

Key Result

Theorem 3.1

If there is a non--trivial solution to tep1--tep2 with eigenvalue $k>0$, then its restriction to $B_R$ is a non--trivial solution to tep3--tep4. Moreover, if there is a non--trivial solution to tep3--tep4 with eigenvalue $k>0$, then it can be extended to $\mathbb{R}^2$ by DtN such that the extension

Figures (1)

  • Figure 1: Visual representations of the three scatterers defined in Table \ref{['scatterers']}.

Theorems & Definitions (17)

  • Theorem 3.1
  • proof
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 7 more