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

By Ali Feizmohammadi, Lauri Oksanen. Journal of Differential Equations, 2019

129 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/75f9c8119a547ace3b0ed9636e4978a06562e130

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Citations in the corpus

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.”
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): “The recent works [10], [29] have introduced a natural and powerful method of higher order linearizations of the nonlinear Dirichlet-to-Neumann map for inverse boundary problems for elliptic PDE, allowing one to solve such problems for nonlinear equations in situations where the corresponding inverse”