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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.17345 |
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| _version_ | 1866908336141631488 |
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| author | Humaikani, Sarah Al Dhia, Anne-Sophie Bonnet-Ben Fliss, Sonia Hazard, Christophe |
| author_facet | Humaikani, Sarah Al Dhia, Anne-Sophie Bonnet-Ben Fliss, Sonia Hazard, Christophe |
| contents | We prove that there are no non-zero square-integrable solutions to a two-dimensional Helmholtz equation in some unbounded inhomogeneous domains which represent junctions of stratified media. More precisely, we consider domains that are unions of three half-planes, where each half-plane is stratified in the direction orthogonal to its boundary. As for the well-known Rellich uniqueness theorem for a homogeneous exterior domain, our result does not require any boundary condition. Our proof is based on half-plane representations of the solution which are derived through a generalization of the Fourier transform adapted to stratified media. A byproduct of our result is the absence of trapped modes at the junction of open waveguides as soon as the angles between branches are greater than $π$/2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Rellich-type theorem for the Helmholtz equation in a junction of stratified media Humaikani, Sarah Al Dhia, Anne-Sophie Bonnet-Ben Fliss, Sonia Hazard, Christophe Analysis of PDEs Spectral Theory We prove that there are no non-zero square-integrable solutions to a two-dimensional Helmholtz equation in some unbounded inhomogeneous domains which represent junctions of stratified media. More precisely, we consider domains that are unions of three half-planes, where each half-plane is stratified in the direction orthogonal to its boundary. As for the well-known Rellich uniqueness theorem for a homogeneous exterior domain, our result does not require any boundary condition. Our proof is based on half-plane representations of the solution which are derived through a generalization of the Fourier transform adapted to stratified media. A byproduct of our result is the absence of trapped modes at the junction of open waveguides as soon as the angles between branches are greater than $π$/2. |
| title | A Rellich-type theorem for the Helmholtz equation in a junction of stratified media |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2504.17345 |