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arXiv · 2012.04435

Quantitative stability of Gel'fand's inverse boundary problem

Dmitri Burago·Sergei Ivanov·Matti Lassas·Jinpeng Lu
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Abstract

In Gel'fand's inverse problem, one aims to determine the topology, differential structure and Riemannian metric of a compact manifold $M$ with boundary from the knowledge of the boundary $\partial M,$ the Neumann eigenvalues $λ_j$ and the boundary values of the eigenfunctions $φ_j|_{\partial M}$. We show that this problem has a stable solution with quantitative stability estimates in a class of manifolds with bounded geometry. More precisely, we show that finitely many eigenvalues and the boundary values of corresponding eigenfunctions, known up to small errors, determine a metric space that is close to the manifold in the Gromov-Hausdorff sense. We provide an algorithm to construct this metric space. This result is based on an explicit estimate on the stability of the unique continuation for the wave operator.

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Dmitri Burago, Sergei Ivanov, Matti Lassas, Jinpeng Lu. 2024-03-25. Quantitative stability of Gel'fand's inverse boundary problem. https://doi.org/10.2140/apde.2025.18.963

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