Inverse problems for elliptic equations with power type nonlinearities
By Matti Lassas, T. Liimatainen, Yi-Hsuan Lin, Mikko Salo. Journal des Mathématiques Pures et Appliquées, 2019
145 citations recorded by the source corpus.
We introduce a method for solving Calderon type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension $2$, and a potential on transversally anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show that one can solve the Calderon problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/d4819295841916ee440f99898e59681121dca86f
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In Applied Mathematics
- Katya Krupchyk
- Gunther Uhlmann
- Yavar Kian
- Ali Feizmohammadi
- C. Carstea
- Mikko Salo
- Plamen Stefanov
- Victor Isakov
In Computational Theory and Mathematics
- Katya Krupchyk
- Gunther Uhlmann
- Ali Feizmohammadi
- Yavar Kian
- C. Carstea
- Mikko Salo
- Plamen Stefanov
- Matti Lassas
In Mathematical Physics
- Katya Krupchyk
- Gunther Uhlmann
- Ali Feizmohammadi
- Yavar Kian
- C. Carstea
- Mikko Salo
- Plamen Stefanov
- Matti Lassas
Citations in the corpus
- A remark on partial data inverse problems for semilinear elliptic equations (2019, methodology): “Similar phenomena for inverse problems for semilinear elliptic PDE have been revealed in the recent works [4], [12].”
- Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities (2019, methodology): “We shall follow [29] and in order to prove this result we shall rely on the implicit function theorem for holomorphic maps between complex Banach spaces, see [38, p.”
- The Calderón inverse problem for isotropic quasilinear conductivities (2021, methodology): “The idea of higher order linearizations was introduced in the context of wave equations in [32] and later adapted to elliptic equations in [13, 35], see also [43, 44] for a second order linearization technique.”