Stability and Lorentzian geometry for an inverse problem of a semilinear wave equation
Lassas, Matti; Liimatainen, Tony; Potenciano-Machado, Leyter; Tyni, Teemu (2025-05-10)
Mathematical Sciences Publishers
10.05.2025
Lassas, M., Liimatainen, T., Potenciano-Machado, L., & Tyni, T. (2025). Stability and Lorentzian geometry for an inverse problem of a semilinear wave equation. Analysis & PDE, 18(5), 1065–1118. https://doi.org/10.2140/apde.2025.18.1065
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© 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open.
https://creativecommons.org/licenses/by/4.0/
© 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open.
https://creativecommons.org/licenses/by/4.0/
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:oulu-202505133318
https://urn.fi/URN:NBN:fi:oulu-202505133318
Tiivistelmä
Abstract
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 q of the nonlinear wave equation □gu+qum=0, m≥4, 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.
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 q of the nonlinear wave equation □gu+qum=0, m≥4, 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.
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- Avoin saatavuus [38506]
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