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

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 −∆ g u + 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.

Inverse problems for real principal type operators

“…Inverse problems for nonlinear hyperbolic equations have been studied in [KLU18, KLOU14, LUW18], and we have also used ideas from related results for elliptic equations [LLLS19,FO19]. We also mention the work [Is91] studying inverse boundary problems for certain general equations of the form P (D)u + V u = 0, where P (D) is a constant coefficient operator.…”

On an inverse boundary value problem for a nonlinear elastic wave equation

“…Our work is still based on the higher order linearization utilized in aforementioned work, but instead of distorted plane waves we will use Gaussian beams. We note here that Gaussian beams have been used to study various inverse problems [2,3,10,11,12,13,18]. We emphasize here that Gaussian beams can be constructed allowing conjugate points.…”

“…The linearization of Λ itself has already been used in [8]. Higher order linearization of Dirichlet-to-Neumann map and the resulted integral identities for semilinear and quasilinear elliptic equations are used [32,17,1,5,23,24,12,19]. Assume u solves (1) with Dirichlet boundary value…”

“…Applying ∂ 2 ∂ǫ 1 ∂ǫ 2 to (1), we obtain the equation for 1) , u (2) ), (t, x) ∈ (0, T ) × Ω, U (12) (t, x) = 0, (t, x) ∈ (0, T ) × ∂Ω, U (12) (0, x) = ∂ ∂t U (12) (0, x) = 0, x ∈ Ω.…”

“…By integration by parts, we get 12) )v + C∇U (12) : ∇v dxdt 1) , u (2) ) v + C∇U (12) : ∇v dxdt 1) , u (2) )v + C∇U (12) : ∇v dxdt 1) , u (2) )v dxdt.…”