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The Cauchy Data and the Scattering Relation

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Geometric Methods in Inverse Problems and PDE Control

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 137))

Abstract

As mentioned in the preface to this volume a combination of unique continuation results with the boundary control method has led to the solution of the inverse problem of determining a metric of a Riemannian manifold (with boundary) from the dynamic Dirichlet-to-Neumann map associated with the wave equation. Although these results are very satisfactory it requires too much information. By just looking at the singularities of the dynamic Dirichlet-to-Neumann (DN) map one can determine the boundary distance function (the minimal travel time along geodesies connecting points on the boundary of a Riemannian manifold) in the case that there are no conjugate points of the metric, i. e. no caustics. This is shown in Section 3 of this article using geometrical optics expansions. A natural question to ask if one can determine the metric from this data alone; this question is at the center of the boundary rigidity problem studied in Riemannian geometry which is one of the main topics of this volume.

The research for this paper was partly supported by NSF and a John Simon Guggenheim fellowship.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, 98195-4350, Seattle, WA, USA

    Gunther Uhlmann

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  1. Gunther Uhlmann
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Editors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, 19104-6395, Philadelphia, PA, USA

    Christopher B. Croke

  2. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    Michael S. Vogelius

  3. Department of Mathematics, University of Washington, 98195-4350, Seattle, WA, USA

    Gunther Uhlmann

  4. Department of Mathematics, University of Virginia, 22903, Charlottesville, VA, USA

    Irena Lasiecka

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Uhlmann, G. (2004). The Cauchy Data and the Scattering Relation. In: Croke, C.B., Vogelius, M.S., Uhlmann, G., Lasiecka, I. (eds) Geometric Methods in Inverse Problems and PDE Control. The IMA Volumes in Mathematics and its Applications, vol 137. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9375-7_10

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  • DOI: https://doi.org/10.1007/978-1-4684-9375-7_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2341-7

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