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Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets

Kian, Y; Kurylev, Y; Lassas, M; Oksanen, L; (2019) Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets. Journal of Differential Equations , 267 (4) pp. 2210-2238. 10.1016/j.jde.2019.03.008. Green open access

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Abstract

We consider a restricted Dirichlet-to-Neumann map Λ S,RT associated with the operator ∂ t2 −Δ g +A+q where Δ g is the Laplace-Beltrami operator of a Riemannian manifold (M,g), and A and q are a vector field and a function on M. The restriction Λ S,RT corresponds to the case where the Dirichlet traces are supported on (0,T)×S and the Neumann traces are restricted on (0,T)×R. Here S and R are open sets, which may be disjoint, on the boundary of M. We show that Λ S,RT determines uniquely, up the natural gauge invariance, the lower order terms A and q in a neighborhood of the set R assuming that R is strictly convex and that the wave equation is exactly controllable from S in time T/2. We give also a global result under a convex foliation condition. The main novelty is the recovery of A and q when the sets R and S are disjoint. We allow A and q to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy.

Type: Article
Title: Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.jde.2019.03.008
Publisher version: https://doi.org/10.1016/j.jde.2019.03.008
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Inverse problems, Wave equation, Partial data
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
URI: https://discovery.ucl.ac.uk/id/eprint/10072538
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