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Stability of the unique continuation for the wave operator via Tataru inequality and applications

Bosi, R; Kurylev, Y; Lassas, M; (2016) Stability of the unique continuation for the wave operator via Tataru inequality and applications. Journal of Differential Equations , 260 (8) pp. 6451-6492. 10.1016/j.jde.2015.12.043. Green open access

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Abstract

In this paper we study the stability of the unique continuation in the case of the wave equation with variable coefficients independent of time. We prove a logarithmic estimate in an arbitrary domain of Rn+1, where all the parameters are calculated explicitly in terms of the C1-norm of the coefficients and on the other geometric properties of the problem. We use the Carleman-type estimate proved by Tataru in 1995 and an iteration of the local stability. We apply the result to the case of a wave equation with data on a cylinder and we get a stable estimate for any positive time, also after the first conjugate point associated with the geodesics of the metric of the variable coefficients.

Type: Article
Title: Stability of the unique continuation for the wave operator via Tataru inequality and applications
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.jde.2015.12.043
Publisher version: http://dx.doi.org/10.1016/j.jde.2015.12.043
Language: English
Additional information: Copyright © 2016. This manuscript version is published under a Creative Commons Attribution Non-commercial Non-derivative 4.0 International licence (CC BY-NC-ND 4.0). This licence allows you to share, copy, distribute and transmit the work for personal and non-commercial use providing author and publisher attribution is clearly stated. Further details about CC BY licences are available at http://creativecommons.org/licenses/by/4.0.
Keywords: Wave equation, Unique continuation property, Stability, Analysis on manifolds, Optimal control time
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1478784
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