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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Time-optimal reconstruction of Riemannian manifold via boundary electromagnetic measurements

Mikhail I. Belishev
  • Saint-Petersburg Department of the Steklov Mathematical Institute, Russian Academy of Sciences, 191023 Saint-Petersburg, Russia.
  • :
/ Maxim N. Demchenko
  • Saint-Petersburg Department of the Steklov Mathematical Institute, Russian Academy of Sciences, 191023 Saint-Petersburg, Russia.
  • :
Published Online: 2011-05-31 | DOI: https://doi.org/10.1515/jiip.2011.028


A dynamical Maxwell system is et = curl h, ht = – curl e in Ω × (0, T), e|t=0 = 0, h|t=0 = 0 in Ω, eθ = ƒ in Ω × [0, T], where Ω is a smooth compact oriented 3-dimensional Riemannian manifold with boundary, (·)θ is a tangent component of a vector at the boundary, e = e ƒ (x, t) and h = h ƒ (x, t) are the electric and magnetic components of the solution. One associates with this system a response operator RT : ƒ ↦ νh ƒ|Ω×(0,T), where ν is an outward normal to Ω.

The time-optimal setup of the inverse problem, which is relevant to the finiteness of the wave speed propagation, is as follows: given R 2T to recover the part ΩT ≔ {x ∈ Ω| dist(x, ∂Ω) < T} of the manifold. As was shown by Belishev, Isakov, Pestov, Sharafutdinov (Doklady Mathematics 61: 353–356, 2000), for T small enough the operator R 2T determines ΩT uniquely up to isometry.

Here we prove that uniqueness holds for arbitrary T > 0 and provide a procedure that recovers ΩT from R 2T. Our approach is a version of the boundary control method (Belishev, 1986).

Keywords.: Maxwell system on manifold; inverse problem; reconstruction of manifold via dynamical response operator; BC-method

Received: 2011-01-30

Published Online: 2011-05-31

Published in Print: 2011-06-01

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 19, Issue 2, Pages 167–188, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip.2011.028, May 2011

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