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

Ali Feizmohammadi; Tony Liimatainen; Yi Hsuan Lin

doi:10.1007/s40818-023-00153-w

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

Department of Applied Mathematics

Research output: Contribution to journal › Article › peer-review

6 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 language	English
Article number	12
Journal	Annals of PDE
Volume	9
Issue number	2
DOIs	https://doi.org/10.1007/s40818-023-00153-w
State	Published - Dec 2023

Keywords

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

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10.1007/s40818-023-00153-w

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title = "An Inverse Problem for a Semilinear Elliptic Equation on Conformally Transversally Anisotropic Manifolds",

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.",

keywords = "Boundary determination, Conformally transversally anisotropic, Gaussian quasimodes, Inverse problems, Riemannian manifold, Semilinear elliptic equation, WKB construction",

author = "Ali Feizmohammadi and Tony Liimatainen and Lin, {Yi Hsuan}",

note = "Publisher Copyright: {\textcopyright} 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.",

year = "2023",

month = dec,

doi = "10.1007/s40818-023-00153-w",

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AU - Feizmohammadi, Ali

AU - Liimatainen, Tony

AU - Lin, Yi Hsuan

PY - 2023/12

Y1 - 2023/12

N2 - 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.

AB - 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.

KW - Boundary determination

KW - Conformally transversally anisotropic

KW - Gaussian quasimodes

KW - Inverse problems

KW - Riemannian manifold

KW - Semilinear elliptic equation

KW - WKB construction

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JF - Annals of PDE

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M1 - 12

ER -

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

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