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arXiv · 2306.05839

Rigidity of inverse problems for nonlinear elliptic equations on manifolds

Ali Feizmohammadi·Yavar Kian·Lauri Oksanen
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Abstract

We consider the inverse problem of determining coefficients appearing in semilinear elliptic equations stated on Riemannian manifolds with boundary given the knowledge of the associated Dirichlet-to-Neumann map. We begin with a negative answer to this problem. Owing to this obstruction, we consider a new formulation of our inverse problem in terms of a rigidity problem. Precisely, we consider cases where the Dirichlet-to-Neumann map of a semilinear equation coincides with the one of a linear equation and ask whether this implies that the equation must indeed be linear. We give a positive answer to this rigidity problem under some assumptions imposed to the Riemannian manifold and to the semilinear term under consideration.

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Ali Feizmohammadi, Yavar Kian, Lauri Oksanen. 2023-06-09. Rigidity of inverse problems for nonlinear elliptic equations on manifolds. https://doi.org/10.1112/blms.13102

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