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Hyperbolic inverse boundary-value problem and time-continuation of the non-stationary Dirichlet-to-Neumann map

Published online by Cambridge University Press:  12 July 2007

Yaroslav Kurylev  and
Matti Lassas
Yaroslav Kurylev
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
Matti Lassas
Affiliation:
Rolf Nevanlinna Institute, University of Helsinki, Helsinki, PO Box 4, FIN-00014, Finland
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Abstract

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Let M be a compact Riemannian manifold with non-empty boundary M. In this paper we consider an inverse problem for the second-order hyperbolic initial-boundary-value problem utt + but + a(x, D)u = 0 in M R+, u|MR+ = f, u|t=0 = ut|t=0 = 0. Our goal is to determine (M, g), b and a(x, D) from the knowledge of the non-stationary Dirichlet-to-Neumann map (the hyperbolic response operator) RT, with sufficiently large T 0. The response operator RT is the map , where is the normal derivative of the solution of the initial-boundary-value problem.

More specifically, we show the following.

  1. (i) It is possible to determine Rt for any t 0 if we know RT for sufficiently large T and some geometric condition upon the geodesic behaviour on (M, g) is satisfied.

  2. (ii) It is then possible to determine (M, g) and b uniquely and the elliptic operator a(x, D) modulo generalized gauge transformations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 132 , Issue 4 , August 2002 , pp. 931 - 949
Copyright
Copyright © Royal Society of Edinburgh 2002