Rigidity of inverse problems for nonlinear elliptic equations on manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916289029603328 |
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| author | Feizmohammadi, Ali Kian, Yavar Oksanen, Lauri |
| author_facet | Feizmohammadi, Ali Kian, Yavar Oksanen, Lauri |
| contents | We consider the inverse problem of determining coefficients appearing in semilinear elliptic equations stated on Riemannian manifolds with boundary given the knowledge of the associated Dirichlet-to-Neumann map. We begin with a negative answer to this problem. Owing to this obstruction, we consider a new formulation of our inverse problem in terms of a rigidity problem. Precisely, we consider cases where the Dirichlet-to-Neumann map of a semilinear equation coincides with the one of a linear equation and ask whether this implies that the equation must indeed be linear. We give a positive answer to this rigidity problem under some assumptions imposed to the Riemannian manifold and to the semilinear term under consideration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_05839 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rigidity of inverse problems for nonlinear elliptic equations on manifolds Feizmohammadi, Ali Kian, Yavar Oksanen, Lauri Analysis of PDEs 35R30, 35J91 We consider the inverse problem of determining coefficients appearing in semilinear elliptic equations stated on Riemannian manifolds with boundary given the knowledge of the associated Dirichlet-to-Neumann map. We begin with a negative answer to this problem. Owing to this obstruction, we consider a new formulation of our inverse problem in terms of a rigidity problem. Precisely, we consider cases where the Dirichlet-to-Neumann map of a semilinear equation coincides with the one of a linear equation and ask whether this implies that the equation must indeed be linear. We give a positive answer to this rigidity problem under some assumptions imposed to the Riemannian manifold and to the semilinear term under consideration. |
| title | Rigidity of inverse problems for nonlinear elliptic equations on manifolds |
| topic | Analysis of PDEs 35R30, 35J91 |
| url | https://arxiv.org/abs/2306.05839 |