On the transmission eigenvalues for scattering by a clamped planar region

TL;DR

The paper introduces a new clamped transmission eigenvalue problem arising from biharmonic scattering in a Kirchhoff-Love plate, formulated on the entire plane $\mathbb{R}^2$. It develops a variational framework via a Helmholtz-modified Helmholtz split, proves discreteness of the transmission eigenvalues in $\mathbb{C}_{+}$ using the analytic Fredholm theorem, and shows that eigenvalues with $k>0$ are recoverable from far-field data. Numerical verification is carried out with boundary-integral methods for unit disk, ellipse, and deformed ellipse, and the far-field based reconstruction is demonstrated with noisy data, highlighting a monotone behavior with respect to the scatterer’s area. The results provide both theoretical and computational foundations for inverse spectral problems in biharmonic exterior scattering and suggest avenues for further study of monotonicity and isoperimetric properties.

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.

Figures (9)

• Figure 1: The graph of the function $|f_{\ell} (\mathrm{i}k)|$ with $k \in [0,5]$ for ${\ell}=0,1,2$ on the left and ${\ell}=3,4,5$ on the right as defined in \ref{['detfn']}.
• Figure 2: Plots of the real part of the eigenfunctions $w$ within $\mathbb{R}^2\backslash \overline{D}$ and $v$ within $D$ for the unit disk corresponding to $k_1\approx 1.61464$(left) and $k_2\approx 3.05164$(right).
• Figure 3: Plots of the real part of the eigenfunctions $w$ within $\mathbb{R}^2\backslash \overline{D}$ and $v$ within $D$ for the unit disk corresponding to $k_3\approx 3.05164$(left) and $k_4\approx 4.36453$(right).
• Figure 4: Plots of the real part of the eigenfunctions $w$ within $\mathbb{R}^2\backslash \overline{D}$ and $v$ within $D$ for the ellipse $(1,0.8)$ corresponding to $k_1\approx 1.81492$(left) and $k_2\approx 3.24382$(right).
• Figure 5: Plots of the real part of the eigenfunctions $w$ within $\mathbb{R}^2\backslash \overline{D}$ and $v$ within $D$ for the ellipse $(1,0.8)$ corresponding to $k_3\approx 3.62554$(left) and $k_4\approx 4.67571$(right).

Theorems & Definitions (8)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1
• proof
• Theorem 4.1
• proof