Inverse problems for semilinear elliptic PDE with a general nonlinearity $a(x,u)$

David Johansson; Janne Nurminen; Mikko Salo

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arXiv · 2312.12196

Inverse problems for semilinear elliptic PDE with a general nonlinearity $a(x,u)$

David Johansson·Janne Nurminen·Mikko Salo

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Abstract

This article studies the inverse problem of recovering a nonlinearity in an elliptic equation $\Delta u + a(x,u) = 0$ from boundary measurements of solutions. Previous results based on first order linearization achieve this under a sign condition on $\partial_u a(x,u)$, and results based on higher order linearization recover the Taylor series of $a(x,u)$ with respect to $u$. We improve these results and show that a general nonlinearity, and not just its Taylor series, is uniquely determined up to gauge near a fixed solution. Our method is based on constructing a good solution map that locally parametrizes solutions of the nonlinear equation by solutions of the linearized equation.

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BibTeX RIS

David Johansson, Janne Nurminen, Mikko Salo. 2023-12-19. Inverse problems for semilinear elliptic PDE with a general nonlinearity $a(x,u)$. https://arxiv.org/abs/2312.12196

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