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Stability and
Lorentzian geometry for an inverse problem of a semilinear
wave equation
Matti Lassas, Tony Liimatainen, Leyter Potenciano-Machado and Teemu Tyni |
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Vol. 18 (2025), No. 5, 1065–1118
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Abstract |
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This paper concerns an inverse boundary value problem for a semilinear wave equation on a globally hyperbolic Lorentzian manifold. We prove a Hölder stability result for recovering an unknown potential of the nonlinear wave equation , , from the Dirichlet-to-Neumann map. Our proof is based on the recent higher-order linearization method and use of Gaussian beams. We also extend earlier uniqueness results by removing the assumptions of convex boundary and that pairs of light-like geodesics can intersect only once. For this, we construct special light-like geodesics and other general constructions in Lorentzian geometry. We expect these constructions to be applicable in studies of related problems as well. |
Keywords
Lorentzian geometry, nonlinear wave equations, inverse
problems
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Mathematical Subject Classification
Primary: 35L71, 58J45
Secondary: 35L05
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Milestones
Received: 30 November 2022
Revised: 17 January 2024
Accepted: 11 April 2024
Published: 10 May 2025
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Authors |
| Matti Lassas | |
| Department of Mathematics and
Statistics University of Helsinki Helsinki Finland |
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| Tony Liimatainen | |
| Department of Mathematics and
Statistics University of Helsinki Helsinki Finland |
Department of Mathematics and
Statistics University of Jyväskylä Jyväskylä Finland |
| Leyter Potenciano-Machado | |
| Department of Mathematics and
Statistics University of Jyväskylä Jyväskylä Finland |
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| Teemu Tyni | |
| Research Unit of Applied and
Computational Mathematics University of Oulu Oulu Finland |
Department of Mathematics and
Statistics University of Helsinki Helsinki Finland |
| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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