Large ∣k∣ Behavior of Complex Geometric Optics Solutions to d‐bar Problems

Complex geometric optics solutions to a system of d‐bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey‐Stewartson II equations are studied for large values of the spectral parameter k. For potentials q∈⋅−2Hsℂ for some s∈1,2, it...

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Published inCommunications on pure and applied mathematics Vol. 76; no. 10; pp. 2221 - 2270
Main Authors Klein, Christian, Sjöstrand, Johannes, Stoilov, Nikola
Format Journal Article
LanguageEnglish
Published Melbourne John Wiley & Sons Australia, Ltd 01.10.2023
John Wiley and Sons, Limited
Wiley
Subjects
Characteristic functions
Electrical impedance
Geometrical optics
Mathematics
Smooth boundaries
1st-order systems
inverse scattering transform
stewartson ii equation
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Summary:Complex geometric optics solutions to a system of d‐bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey‐Stewartson II equations are studied for large values of the spectral parameter k. For potentials q∈⋅−2Hsℂ for some s∈1,2, it is shown that the solution converges as the geometric series in 1/ks−1. For potentials q being the characteristic function of a strictly convex open set with smooth boundary, this still holds with s = 3/2, i.e., with 1/∣k∣ instead of 1/ks−1. The leading‐order contributions are computed explicitly. Numerical simulations show the applicability of the asymptotic formulae for the example of the characteristic function of the disk. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.
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ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.22075