Feizmohammadi, A;
Oksanen, L;
(2020)
An inverse problem for a semi-linear elliptic equation in Riemannian geometries.
Journal of Differential Equations
, 269
(6)
pp. 4683-4719.
10.1016/j.jde.2020.03.037.
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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 BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10095828 |
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