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February  2015, 9(1): 127-141. doi: 10.3934/ipi.2015.9.127

## Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type

 1 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin 2 Weierstrass Institute, Mohrenstr. 39, 10117 Berlin 3 Graduate School of Mathematical Sciences University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914

Received  June 2014 Revised  September 2014 Published  January 2015

Consider the two-dimensional inverse elastic scattering problem of recovering a piecewise linear rigid rough or periodic surface of rectangular type for which the neighboring line segments are always perpendicular. We prove the global uniqueness with at most two incident elastic plane waves by using near-field data. If the Lamé constants satisfy a certain condition, then the data of a single plane wave is sufficient to imply the uniqueness. Our proof is based on a transcendental equation for the Navier equation, which is derived from the expansion of analytic solutions to the Helmholtz equation. The uniqueness results apply also to an inverse scattering problem for non-convex bounded rigid bodies of rectangular type.
Citation: Johannes Elschner, Guanghui Hu, Masahiro Yamamoto. Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type. Inverse Problems & Imaging, 2015, 9 (1) : 127-141. doi: 10.3934/ipi.2015.9.127
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##### References:
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