Reconstruction and stability in Gelfand’s inverse interior spectral problem

Bosi, Roberta; Kurylev, Yaroslav; Lassas, Matti

doi:10.2140/apde.2022.15.273

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Reconstruction and stability in Gelfand's inverse interior spectral problem

Roberta Bosi, Yaroslav Kurylev and Matti Lassas

Vol. 15 (2022), No. 2, 273–326

DOI: 10.2140/apde.2022.15.273

Abstract

Assume that $M$ 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 $Δ_{g}$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$ . We then construct a stable approximation to the manifold $(M, g)$ . Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from $M$ when the above data are given with a small error. We give an explicit $\log$ - $\log$ -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.

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Keywords

inverse spectral problems, Riemannian manifolds

Mathematical Subject Classification 2010

Primary: 35R30, 58J05

Milestones

Received: 4 October 2018

Revised: 3 August 2020

Accepted: 6 October 2020

Published: 12 April 2022

Authors

Roberta Bosi
Department of Mathematics and Statistics University of Helsinki Helsinki Finland

Yaroslav Kurylev
Department of Mathematics University College London London United Kingdom

Matti Lassas
Department of Mathematics and Statistics University of Helsinki Helsinki Finland

Vol. 15, No. 2, 2022