Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary

Oleg Yu. Imanuvilov; Masahiro Yamamoto

doi:10.3934/ipi.2019054

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2019, Volume 13, Issue 6: 1213-1258. Doi: 10.3934/ipi.2019054

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Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary

Oleg Yu. Imanuvilov^1, and
Masahiro Yamamoto^2,3,4, ,

1.
Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, CO 80523-1874, USA
2.
Department of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153, Japan
3.
Honorary Member of Academy of Romanian Scientists, Splaiul Independentei Street, no 54, 050094 Bucharest Romania
4.
Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

^* Corresponding author: Masairo Yamamoto
^* Corresponding author: Masairo Yamamoto

dedicated to the memory of Professor Yaroslav Kurylev

Received: August 31, 2018

Revised: March 31, 2019

Published: October 2019

The first author is supported by NSF grant DMS 13129000.
The second author is supported by Grant-in-Aid for Scientific Research (S) 15H05740 of Japan Society for the Promotion of Science and by The National Natural Science Foundation of China (no. 11771270, 91730303), and prepared with the support of the "RUDN University Program 5-100"

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Abstract

In this paper, we establish conditional stability estimates for two inverse problems of determining metrics in two dimensional Laplace-Beltrami operators. As data, in the first inverse problem we adopt spectral data on an arbitrarily fixed subboundary, while in the second, we choose the Dirichlet-to-Neumann map limited on an arbitrarily fixed subboundary. The conditional stability estimates for the two inverse problems are stated as follows. If the difference between spectral data or Dirichlet-to-Neumann maps related to two metrics $ {\bf{g}}_1 $ and $ {\bf{g}}_2 $ is small, then $ {\bf{g}}_1 $ and $ {\bf{g}}_2 $ are close in $ L^2(\Omega) $ modulo a suitable diffeomorphism within a priori bounds of $ {\bf{g}}_1 $ and $ {\bf{g}}_2 $. Both stability estimates are of the same double logarithmic rate.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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