Inverse Scattering for Obstacles and Related Problems

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The proposers will work on inverse coefficient problems where the given data is the scattering amplitude at fixed energy. The desired result is that this determines the unknown coefficients. Specifically,they will consider this problem for the acoustic wave equation inan exterior domain with the sound speed and the boundary of the domain as unknowns to be determined by the scattering data. They arealso interested in the related problem of recovering metrics modulodiffeomorphisms from scattering data for Laplace-Beltrami operator.The proposed method of attack is via integral equations that they derived from Faddeev-type Green's functions in earlier work. The preceding problems will be posed in three or more dimensions. In twodimensions this approach needs to be combined with other methods likethe d-bar equation. The proposers will investigate ways of doing thisfor the wave equation in an anisotropic medium. Inverse scattering problems arise when one wishes to recover physicalproperties of an object from data obtained by remote observations.Interpreted broadly, they include methods oil exploration and medicalimaging. The problems that will be studied in this project include showing that it is theoretically possible to determine both the acousticproperties of a medium surrounding a solid obstacle and the shape of the obstacle itself from reflected sound waves.