Inverse problems for elliptic equations with power type nonlinearities
Lassas, M., Liimatainen, T., Lin, Y.-H., & Salo, M. (2021). Inverse problems for elliptic equations with power type nonlinearities. Journal de Mathematiques Pures et Appliquees, 145, 44-82. https://doi.org/10.1016/j.matpur.2020.11.006
Published inJournal de Mathematiques Pures et Appliquees
DisciplineInversio-ongelmien huippuyksikköMatematiikkaCentre of Excellence in Inverse ProblemsMathematics
Embargoed until: 2023-02-01Request copy from author
© 2020 Elsevier
We introduce a method for solving Calderón 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 . In the Euclidean case, we show that one can solve the Calderón problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions.
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Related funder(s)Academy of Finland
Funding program(s)Academy Project, AoF; Centre of Excellence, AoF
Additional information about fundingAll authors were supported by the Finnish Centre of Excellence in Inverse Modelling and Imaging (Academy of Finland grant 284715). M.S. was also supported by the Academy of Finland (grant 309963) and by the European Research Council under Horizon 2020 (ERC CoG 770924). Y.-H. L. is partially supported by the Ministry of Science and Technology, Taiwan, under the Columbus Pro-gram:MOST-109-2636-M-009-006, 2020-2025. ...
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