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arXiv · 1904.00608

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

Ali Feizmohammadi·Lauri Oksanen
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Abstract

We study the inverse problem of unique recovery of a complex-valued scalar function $V:\mathcal M \times \mathbb C\to \mathbb C$, defined over a smooth compact Riemannian manifold $(\mathcal M,g)$ with smooth boundary, given the Dirichlet to Neumann map, in a suitable sense, for the elliptic semi-linear equation $-\Delta_{g}u+V(x,u)=0$. We show that under some geometrical assumptions uniqueness can be proved for a large class of non-linearities. 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 non-linear interactions result in the study of a weighted transform along geodesics, that we call the Jacobi weighted ray transform.

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Ali Feizmohammadi, Lauri Oksanen. 2019-04-01. An inverse problem for a semi-linear elliptic equation in Riemannian geometries. https://doi.org/10.1016/j.jde.2020.03.037

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