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

Ali Feizmohammadi*, Tony Liimatainen, Yi Hsuan Lin

*此作品的通信作者

研究成果: Article同行評審

3 引文 斯高帕斯(Scopus)

摘要

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.

原文English
文章編號12
期刊Annals of PDE
9
發行號2
DOIs
出版狀態Published - 12月 2023

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