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arXiv · 2504.17345

A Rellich-type theorem for the Helmholtz equation in a junction of stratified media

Sarah Al Humaikani·Anne-Sophie Bonnet-Ben Dhia·Sonia Fliss·Christophe Hazard
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Abstract

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 $\pi$/2.

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Sarah Al Humaikani, Anne-Sophie Bonnet-Ben Dhia, Sonia Fliss, Christophe Hazard. 2025-04-24. A Rellich-type theorem for the Helmholtz equation in a junction of stratified media. https://arxiv.org/abs/2504.17345

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