Rigidity in the Lorentzian Calderón problem with formally determined data
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910623173967872 |
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| author | Oksanen, Lauri Rakesh Salo, Mikko |
| author_facet | Oksanen, Lauri Rakesh Salo, Mikko |
| contents | We study the Lorentzian Calderón problem, where the objective is to determine a globally hyperbolic Lorentzian metric up to a boundary fixing diffeomorphism from boundary measurements given by the hyperbolic Dirichlet-to-Neumann map. This problem is a wave equation analogue of the Calderón problem on Riemannian manifolds. We prove that if a globally hyperbolic metric agrees with the Minkowski metric outside a compact set and has the same hyperbolic Dirichlet-to-Neumann map as the Minkowski metric, then it must be the Minkowski metric up to diffeomorphism. In fact we prove the same result with a much smaller amount of measurements, thus solving a formally determined inverse problem. To prove these results we introduce a new method for geometric hyperbolic inverse problems. The method is based on distorted plane wave solutions and on a combination of geometric, topological and unique continuation arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18604 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity in the Lorentzian Calderón problem with formally determined data Oksanen, Lauri Rakesh Salo, Mikko Analysis of PDEs Mathematical Physics Differential Geometry We study the Lorentzian Calderón problem, where the objective is to determine a globally hyperbolic Lorentzian metric up to a boundary fixing diffeomorphism from boundary measurements given by the hyperbolic Dirichlet-to-Neumann map. This problem is a wave equation analogue of the Calderón problem on Riemannian manifolds. We prove that if a globally hyperbolic metric agrees with the Minkowski metric outside a compact set and has the same hyperbolic Dirichlet-to-Neumann map as the Minkowski metric, then it must be the Minkowski metric up to diffeomorphism. In fact we prove the same result with a much smaller amount of measurements, thus solving a formally determined inverse problem. To prove these results we introduce a new method for geometric hyperbolic inverse problems. The method is based on distorted plane wave solutions and on a combination of geometric, topological and unique continuation arguments. |
| title | Rigidity in the Lorentzian Calderón problem with formally determined data |
| topic | Analysis of PDEs Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2409.18604 |