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arXiv · 2112.08305

An inverse problem for a semilinear elliptic equation on conformally transversally anisotropic manifolds

Ali Feizmohammadi·Tony Liimatainen·Yi-Hsuan Lin
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Abstract

Given a conformally transversally anisotropic manifold $(M,g)$, we consider the semilinear elliptic equation $$(-\Delta_{g}+V)u+qu^2=0\quad \text{on $M$}.$$ 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 semilinear elliptic equation. This extends the previously known results of the works [FO20, LLLS21a]. Our proof is based on analyzing higher order linearizations of the semilinear equation with non-vanishing boundary traces and also the study of interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation.

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Ali Feizmohammadi, Tony Liimatainen, Yi-Hsuan Lin. 2021-12-15. An inverse problem for a semilinear elliptic equation on conformally transversally anisotropic manifolds. https://doi.org/10.1007/s40818-023-00153-w

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