An Inverse Problem for a Semilinear Elliptic Equation on Conformally Transversally Anisotropic Manifolds

Ali Feizmohammadi*, Tony Liimatainen, Yi Hsuan Lin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

Given a conformally transversally anisotropic manifold (M, g), we consider the semilinear elliptic equation (-Δg+V)u+qu2=0onM. We show that an a priori unknown smooth function q can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the equation. This extends the previously known results of the works Feizmohammadi and Oksanen (J Differ Equ 269(6):4683–4719, 2020), Lassas et al. (J Math Pures Appl 145:44–82, 2021). Our proof is based on over-differentiating the equation: We linearize the equation to orders higher than the order two of the nonlinearity qu2 , and introduce non-vanishing boundary traces for the linearizations. We study interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation. We develop an asymptotic calculus to solve Laplace equations, which have these interactions as source terms.

Original languageEnglish
Article number12
JournalAnnals of PDE
Volume9
Issue number2
DOIs
StatePublished - Dec 2023

Keywords

  • Boundary determination
  • Conformally transversally anisotropic
  • Gaussian quasimodes
  • Inverse problems
  • Riemannian manifold
  • Semilinear elliptic equation
  • WKB construction

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