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Reconstruction and
stability in Gelfand's inverse interior spectral problem
Roberta Bosi, Yaroslav Kurylev and Matti Lassas |
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Vol. 15 (2022), No. 2, 273–326
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Abstract |
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Assume that is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian on as well as the corresponding eigenfunctions restricted on an open set in . We then construct a stable approximation to the manifold . Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from when the above data are given with a small error. We give an explicit --type stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gelfand inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric boundary control method. |
Keywords
inverse spectral problems, Riemannian manifolds
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Mathematical Subject Classification 2010
Primary: 35R30, 58J05
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Milestones
Received: 4 October 2018
Revised: 3 August 2020
Accepted: 6 October 2020
Published: 12 April 2022
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Authors |
| Roberta Bosi | |
| Department of Mathematics and
Statistics University of Helsinki Helsinki Finland |
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| Yaroslav Kurylev | |
| Department of Mathematics University College London London United Kingdom |
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| Matti Lassas | |
| Department of Mathematics and
Statistics University of Helsinki Helsinki Finland |
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