2019
DOI: 10.48550/arxiv.1904.00608
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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 e… Show more

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Cited by 16 publications
(29 citation statements)
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“…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.…”
Section: Previous Literature and Outlookmentioning
confidence: 99%
“…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.…”
Section: Previous Literature and Outlookmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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…”
Section: Introductionmentioning
confidence: 99%
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