Stability estimates for inverse problems for semi-linear wave equations on Lorentzian manifolds
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866918017769668608 |
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| author | Lassas, Matti Liimatainen, Tony Potenciano-Machado, Leyter Tyni, Teemu |
| author_facet | Lassas, Matti Liimatainen, Tony Potenciano-Machado, Leyter Tyni, Teemu |
| contents | This paper concerns an inverse boundary value problem of recovering a zeroth order time-dependent term of a semi-linear wave equation on a globally hyperbolic Lorentzian manifold. We show that an unknown potential $q$ in the non-linear wave equation $\square_g u +q u^m=0$, $m\geq 4$, can be recovered in a Hölder stable way from the Dirichlet-to-Neumann map. Our proof is based on the higher order linearization method and the use of Gaussian beams. Unlike some related works, we do not assume that the boundary is convex or that pairs of lightlike geodesics can intersect only once. For this, we introduce some general constructions in Lorentzian geometry. We expect these constructions to be applicable to studies of related problems as well. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_12257 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Stability estimates for inverse problems for semi-linear wave equations on Lorentzian manifolds Lassas, Matti Liimatainen, Tony Potenciano-Machado, Leyter Tyni, Teemu Analysis of PDEs 58J45, 35L71, 35L05 This paper concerns an inverse boundary value problem of recovering a zeroth order time-dependent term of a semi-linear wave equation on a globally hyperbolic Lorentzian manifold. We show that an unknown potential $q$ in the non-linear wave equation $\square_g u +q u^m=0$, $m\geq 4$, can be recovered in a Hölder stable way from the Dirichlet-to-Neumann map. Our proof is based on the higher order linearization method and the use of Gaussian beams. Unlike some related works, we do not assume that the boundary is convex or that pairs of lightlike geodesics can intersect only once. For this, we introduce some general constructions in Lorentzian geometry. We expect these constructions to be applicable to studies of related problems as well. |
| title | Stability estimates for inverse problems for semi-linear wave equations on Lorentzian manifolds |
| topic | Analysis of PDEs 58J45, 35L71, 35L05 |
| url | https://arxiv.org/abs/2106.12257 |