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An Inverse Scattering Problem

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An Introduction to the Mathematical Theory of Inverse Problems

Part of the book series: Applied Mathematical Sciences ((AMS,volume 120))

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Abstract

We consider acoustic waves that travel in a medium such as a fluid. Let v(x, t) be the velocity vector of a particle at \(x \in {\mathbb{R}}^{3}\) and time t. Let p(x, t), ρ(x, t), and S(x, t) denote the pressure, density, and specific entropy, respectively, of the fluid. We assume that no exterior forces act on the fluid.

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Notes

  1. 1.

    One has to modify the proof in [49] where n ∈ C(Ω) is assumed.

  2. 2.

    As proper assumptions on v, w one requires v, w ∈ L 2(D) and v − w ∈ H 2(D). The differential equations are understood in the “ultra weak sense”; see [143].

  3. 3.

    Actually, it was a scattering problem for which the Factorization Method was first discovered ([141]) before it was applied to the problem of electrical impedance tomography in [22, 23].

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  1. Department of Mathematics, Karlsruhe Institute of Technology (KIT), Kaiserstrasse 89, 76133, Karlsruhe, Germany

    Andreas Kirsch

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  1. Andreas Kirsch
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Kirsch, A. (2011). An Inverse Scattering Problem. In: An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences, vol 120. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8474-6_6

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