On the transmission eigenvalues for scattering by a clamped planar region

Isaac Harris; Andreas Kleefeld; Heejin Lee

doi:10.3934/ipi.2025038

Article Contents

2026, Volume 21: 152-172. Doi: 10.3934/ipi.2025038

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On the transmission eigenvalues for scattering by a clamped planar region

1.
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
2.
Forschungszentrum Jülich GmbH, Jülich Supercomputing Centre, Wilhelm-Johnen-Straße, 52425 Jülich, Germany
3.
University of Applied Sciences Aachen, Faculty of Medical Engineering, and Technomathematics, Heinrich-Mußmann-Str. 1, 52428 Jülich, Germany
4.
Zu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University, Kunshan, 215316, Jiangsu Province, China

^*Corresponding author: Isaac Harris
^*Corresponding author: Isaac Harris

Received: February 2025

Revised: August 2025

Early access: August 26, 2025

Published online: August 26, 2025

The author I. Harris is supported in part by the NSF DMS grant [2208256].

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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.

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Mathematics Subject Classification: 35P25, 35J30.

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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 (22)

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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)

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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)

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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)

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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)

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Figure 6. Plots of the real part of the eigenfunctions $ w $ within $ \mathbb{R}^2\backslash \overline{D} $ and $ v $ within $ D $ for the deformed ellipse with $ \epsilon = 0.3 $ corresponding to $ k_1\approx 1.91665 $(left) and $ k_2\approx 3.38373 $(right)

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Figure 7. Plots of the real part of the eigenfunctions $ w $ within $ \mathbb{R}^2\backslash \overline{D} $ and $ v $ within $ D $ for the deformed ellipse with $ \epsilon = 0.3 $ corresponding to $ k_3\approx 3.75151 $(left) and $ k_4\approx 4.96416 $(right)

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Figure 8. The recovered transmission eigenvalues for the unit disk from the far field data. Recall that the first three distinct eigenvalues are given by $ k\approx1.61464, \, 3.05164, \, 4.36453 $ from Table 1

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Figure 9. The recovered transmission eigenvalues for the peanut–shaped region from the far field data. Left: peanut–shaped scattering region; Right: reconstruction of the transmission eigenvalues. The reconstructed transmission eigenvalues are given by $ k\approx $ 2.18121, 3.4698, 4.78523, and $ 4.91946 $ which are approximately the values given in Table 4

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Table 1. The first five transmission eigenvalues (TE) for a unit disk via the determinant with 20 digits accuracy using various $ \ell \in\mathbb{N}\cup \{ 0\} $ and the boundary element collocation method (BEM) with 120 collocation nodes

TE	determinant	BEM
$ k_1 $	1.6146349995639885158 $ (\ell=0) $	1.61464
$ k_2 $	3.0516335028155405705 $ (\ell=1) $	3.05164
$ k_3 $	3.0516335028155405705 $ (\ell=1) $	3.05164
$ k_4 $	4.3645169097857215923 $ (\ell=2) $	4.36453
$ k_5 $	4.3645169097857215923 $ (\ell=2) $	4.36453

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Table 2. The first four transmission eigenvalues (TE) for various ellipses with half-axis $ (a, b) $ using the boundary element collocation method (BEM) with 120 collocation nodes

TE	$ (1, 1) $	$ (1, 0.9) $	$ (1, 0.8) $	$ (1, 0.7) $	$ (1, 0.6) $	$ (1, 0.5) $
$ k_1 $	1.61464	1.70401	1.81492	1.95646	2.14377	2.40418
$ k_2 $	3.05164	3.13673	3.24382	3.38233	3.56787	3.82845
$ k_3 $	3.05164	3.30629	3.62554	4.03737	4.58853	5.26231
$ k_4 $	4.36453	4.54060	4.67571	4.82125	5.00599	5.36324

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Table 3. The first four transmission eigenvalues (TE) for various deformed ellipses with parameter $ \epsilon $ using the boundary element collocation method (BEM) with 120 collocation nodes

TE	$ \epsilon=0.1 $	$ \epsilon=0.2 $	$ \epsilon=0.3 $
$ k_1 $	1.88515	1.89716	1.91665
$ k_2 $	3.31667	3.34174	3.38373
$ k_3 $	3.80574	3.77859	3.75151
$ k_4 $	4.77845	4.86338	4.96416

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Table 4. The first four transmission eigenvalues (TE) for the peanut–shaped region using the boundary element collocation method (BEM) with 120 collocation nodes

TE
$ k_1 $	2.13093
$ k_2 $	3.41900
$ k_3 $	4.70289
$ k_4 $	4.89266

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