Uniqueness to inverse acoustic and elastic medium scattering problems with hyper-singular source method

Abstract: This paper is concerned with inverse scattering problems of determining the support of an isotropic and homogeneous penetrable body from knowledge of multi-static far-field patterns in acoustics and in linear elasticity. The normal derivative of the total fields admits no jump on the interface of the scatterer in the trace sense. If the contrast function of the refractive index function or the density function has a positive lower bound near the boundary, we propose a hyper-singular source method to prove uniqueness of inverse scattering with all incoming plane waves at a fixed energy. It is based on subtle analysis on the leading part of the scattered field when hyper-singular sources caused by the first derivative of the fundamental solution approach to a boundary point. As a by-product, we show that this hyper-singular method can be also used to determine the boundary value of a Holder continuous refractive index function in acoustics or a Holder continuous density function in linear elasticity.