Helmholtz transmission problem and intrinsic impedance scattering problem on extension domains
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908779676696576 |
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| author | Claret, Gabriel |
| author_facet | Claret, Gabriel |
| contents | We consider a transmission problem for the Helmholtz equation across the boundary of an extension domain. A such boundary can be Lipschitz, fractal, or of varying Hausdorff dimension for instance. We generalise the notions of layer potential and Neumann-Poincar{é} operators, and of Calder{ó}n projectors in that context. Those boundary operators allow to connect the transmission problem (on the whole space) to one-sided problems -- notably, scattering problems -- with Dirichlet, Neumann and Robin boundary conditions. Since an extension domain needs no specific boundary measure, the Robin (impedance) condition is not understood in a boundary L^2-type space, rather by duality on the trace space itself. We discuss the well-posedness of the impedance scattering problem in that framework and compare to the classical L^2 setting. Our analysis allows to generalise optimisation results for acoustic scattering when the obstacle is an extension domain in any dimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_14866 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Helmholtz transmission problem and intrinsic impedance scattering problem on extension domains Claret, Gabriel Analysis of PDEs Mathematical Physics Functional Analysis Optimization and Control We consider a transmission problem for the Helmholtz equation across the boundary of an extension domain. A such boundary can be Lipschitz, fractal, or of varying Hausdorff dimension for instance. We generalise the notions of layer potential and Neumann-Poincar{é} operators, and of Calder{ó}n projectors in that context. Those boundary operators allow to connect the transmission problem (on the whole space) to one-sided problems -- notably, scattering problems -- with Dirichlet, Neumann and Robin boundary conditions. Since an extension domain needs no specific boundary measure, the Robin (impedance) condition is not understood in a boundary L^2-type space, rather by duality on the trace space itself. We discuss the well-posedness of the impedance scattering problem in that framework and compare to the classical L^2 setting. Our analysis allows to generalise optimisation results for acoustic scattering when the obstacle is an extension domain in any dimension. |
| title | Helmholtz transmission problem and intrinsic impedance scattering problem on extension domains |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis Optimization and Control |
| url | https://arxiv.org/abs/2601.14866 |