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Quantitative
stability of Gel'fand's inverse boundary problem
Dmitri Burago, Sergei Ivanov, Matti Lassas and Jinpeng Lu |
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Vol. 18 (2025), No. 4, 963–1035
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Abstract |
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In Gel’fand’s inverse problem, one aims to determine the topology, differential structure and Riemannian metric of a compact manifold with boundary from the knowledge of the boundary , the Neumann eigenvalues and the boundary values of the eigenfunctions . 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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Dedicated to the memory of Yaroslav Kurylev |
Keywords
Gel'fand's inverse problem, stability, quantitative unique
continuation, wave operator, boundary control method
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Mathematical Subject Classification
Primary: 35R30, 58J50, 53C21, 58J45
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Milestones
Received: 17 February 2023
Revised: 5 December 2023
Accepted: 20 March 2024
Published: 27 March 2025
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Authors |
| Dmitri Burago | |
| Department of Mathematics Pennsylvania State University University Park, PA United States |
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| Sergei Ivanov | |
| St. Petersburg Department of Steklov
Mathematical Institute Russian Academy of Sciences St. Petersburg Russia |
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| Matti Lassas | |
| Department of Mathematics and
Statistics University of Helsinki Helsinki Finland |
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| Jinpeng Lu | |
| Department of Mathematics and
Statistics University of Helsinki Helsinki Finland |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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