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An inverse problem for a semi-linear elliptic equation in Riemannian geometries

Feizmohammadi, A; Oksanen, L; (2020) An inverse problem for a semi-linear elliptic equation in Riemannian geometries. Journal of Differential Equations 10.1016/j.jde.2020.03.037. (In press). Green open access

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Abstract

We study the inverse problem of unique recovery of a complex-valued scalar function V : M × C → C, defined over a smooth compact Riemannian manifold (M, g) with smooth boundary, given the Dirichlet-to-Neumann map, in a suitable sense, for the elliptic semi-linear equation −∆gu + V (x, u) = 0. We show that uniqueness holds for a large class of non-linearities when the manifold is conformally transversally anisotropic. The proof is constructive and is based on a multiple-fold linearization of the semi-linear equation near complex geometric optic solutions for the linearized operator and the resulting non-linear interactions. These interactions result in the study of a weighted integral transform along geodesics, that we call the Jacobi weighted ray transform.

Type: Article
Title: An inverse problem for a semi-linear elliptic equation in Riemannian geometries
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.jde.2020.03.037
Publisher version: https://doi.org/10.1016/j.jde.2020.03.037
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10095828
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