The anisotropic interior transmission eigenvalue problem with a conductive boundary

Victor Hughes; Isaac Harris; Jiguang Sun

doi:10.3934/cac.2025007

Article Contents

June 2025, Volume 4, 60-82. Doi: 10.3934/cac.2025007

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The anisotropic interior transmission eigenvalue problem with a conductive boundary

1.
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
2.
Department of Mathematics, Michigan Technological University, Houghton, MI 49931, USA

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

Received: April 2025

Published online: May 6, 2025

The research of I. Harris and V. Hughes is partially supported by the [NSF DMS Grant 2107891].

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Abstract

In this paper, we study the transmission eigenvalue problem for an anisotropic material with a conductive boundary. We prove that the transmission eigenvalues for this problem exist and are, at most, a discrete set. We also study the dependence of the transmission eigenvalues on the physical parameters and prove that the first transmission eigenvalue is monotonic. We then consider the limiting behavior of the transmission eigenvalues as the conductive boundary parameter $ \eta $ vanishes or goes to infinity in magnitude. Finally, we provide numerical examples on three domains to demonstrate our theoretical results.

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Mathematics Subject Classification: Primary: 35P25, 35J30; Secondary: 65N30.

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Figure 1. The unit circle, unit square and L–Shaped domains

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Table 1. Convergence with respect to $ \eta \to 0 $ for the unit disk ($ A = 0.4 I $ and $ n = 3 $ for $ \eta>0 $; $ A = 3 I $ and $ n = 0.7 $ for $ \eta<0 $)

$ \eta $	$ k_{\eta} $	EOC	$ \eta $	$ k_\eta $	EOC
1	1.6010	N/A	-1	5.8032	N/A
1/2	1.7353	1.7003	-1/2	5.8143	0.9223
1/4	1.7697	1.2386	-1/4	5.8203	0.9542
1/8	1.7832	1.1004	-1/8	5.8234	0.9556
1/16	1.7893	1.0498	-1/16	5.8250	0.9569
1/32	1.7922	1.0255	-1/32	5.8258	0.9175
1/64	1.7936	1	-1/64	5.8262	0.8480
1/128	1.7943	1	-1/128	5.8264	0.7370
1/256	1.7946	1	-1/256	5.8265	0.5850
1/512	1.7948	1	-1/512	5.8266	1

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Table 2. Convergence with respect to $ \eta \to \pm \infty $ for the unit disk. ($ A = 0.4 I $ and $ n = 3 $ for $ \eta>0 $ and $ A = 3 I $ and $ n = 0.7 $ for $ \eta<0 $)

$ \eta $	$ k_{\eta} $	EOC	$ \eta $	$ k_\eta $	EOC
1	1.6010	N/A	-1	5.8032	N/A
2	1.1415	1.4565	-2	5.7839	0.00822
4	0.9828	1.3310	-4	5.7537	0.01295
8	0.9258	1.1342	-8	5.7124	0.0179
16	0.9010	1.0586	-16	2.5944	4.1248
32	0.8893	1.0318	-32	2.4881	1.1866
64	0.8837	1	-64	2.4443	1.0765
128	0.8809	1	-128	2.4241	1.0333
256	0.8795	1	-256	2.4143	1.0266
512	0.8788	1	-512	2.4095	1.0153

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Table 3. Monotonicity with respect to $ A = aI $ for the unit square. Here, we take $ n = 0.75 $ and $ \eta = -2 $ for $ a>1 $ where as $ n = 2 $ and $ \eta = 2 $ for $ a<1 $

$ a $	$ k_{1}(a) $	$ a $	$ k_{1}(a) $
$ 1.5 $	$ 18.3370 $	$ 0.3 $	$ 3.3113 $
$ 2 $	$ 14.1549 $	$ 0.4 $	$ 4.2377 $
$ 2.5 $	$ 11.8953 $	$ 0.5 $	$ 5.4361 $
$ 3 $	$ 10.7253 $	$ 0.6 $	$ 6.5176 $
$ 3.5 $	$ 10.2221 $	$ 0.7 $	$ 7.6965 $
$ 4 $	$ 9.9292 $	$ 0.8 $	$ 9.2953 $

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Table 4. Monotonicity with respect to $ n $ for the L–Shaped domain. Here, we take $ A = 0.7 I $ and $ \eta = 2 $ for $ n>1 $ where as $ A = 3 I $ and $ \eta = -2 $ for $ n<1 $

$ n $	$ k_{1}(n) $	$ n $	$ k_1(n) $
$ 1.5 $	$ 12.9868 $	$ 0.3 $	$ 10.5159 $
$ 2 $	$ 9.0795 $	$ 0.4 $	$ 11.0014 $
$ 2.5 $	$ 7.2272 $	$ 0.5 $	$ 11.5661 $
$ 3 $	$ 6.0632 $	$ 0.6 $	$ 12.2386 $
$ 3.5 $	$ 5.3701 $	$ 0.7 $	$ 12.9390 $
$ 4 $	$ 4.8812 $	$ 0.8 $	$ 13.5370 $

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Table 5. Convergence with respect to $ \eta \to \pm \infty $ for the L-shaped domain ($ A = 0.4 I $ and $ n = 3 $ for $ \eta>0 $ and $ A = 3 I $ and $ n = 0.7 $ for $ \eta<0 $)

$ \eta $	$ k_{\eta} $	$ \eta $	$ k_\eta $
$ 1 $	$ 3.9995 $	$ -1 $	$ 13.0503 $
$ 2 $	$ 3.8281 $	$ -2 $	$ 12.9390 $
$ 4 $	$ 3.4828 $	$ -4 $	$ 12.6623 $
$ 8 $	$ 2.7805 $	$ -8 $	$ 11.7693 $
$ 16 $	$ 2.4898 $	$ -16 $	$ 10.1467 $
$ 32 $	$ 2.3732 $	$ -32 $	$ 7.2242 $
$ 64 $	$ 2.3196 $	$ -64 $	$ 6.6044 $
$ 128 $	$ 2.2936 $	$ -128 $	$ 6.3938 $
$ 256 $	$ 2.2807 $	$ -256 $	$ 6.3000 $
$ 512 $	$ 2.2742 $	$ -512 $	$ 6.2549 $

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