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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.12196 |
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| _version_ | 1866914537959063552 |
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| author | Johansson, David Nurminen, Janne Salo, Mikko |
| author_facet | Johansson, David Nurminen, Janne Salo, Mikko |
| contents | This article studies the inverse problem of recovering a nonlinearity in an elliptic equation $Δu + a(x,u) = 0$ from boundary measurements of solutions. Previous results based on first order linearization achieve this under a sign condition on $\partial_u a(x,u)$, and results based on higher order linearization recover the Taylor series of $a(x,u)$ with respect to $u$. We improve these results and show that a general nonlinearity, and not just its Taylor series, is uniquely determined up to gauge near a fixed solution. Our method is based on constructing a good solution map that locally parametrizes solutions of the nonlinear equation by solutions of the linearized equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12196 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Inverse problems for semilinear elliptic PDE with a general nonlinearity $a(x,u)$ Johansson, David Nurminen, Janne Salo, Mikko Analysis of PDEs This article studies the inverse problem of recovering a nonlinearity in an elliptic equation $Δu + a(x,u) = 0$ from boundary measurements of solutions. Previous results based on first order linearization achieve this under a sign condition on $\partial_u a(x,u)$, and results based on higher order linearization recover the Taylor series of $a(x,u)$ with respect to $u$. We improve these results and show that a general nonlinearity, and not just its Taylor series, is uniquely determined up to gauge near a fixed solution. Our method is based on constructing a good solution map that locally parametrizes solutions of the nonlinear equation by solutions of the linearized equation. |
| title | Inverse problems for semilinear elliptic PDE with a general nonlinearity $a(x,u)$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2312.12196 |