On the Uniqueness of a Solution to an Inverse Problem of Scattering by an Inhomogeneous Solid with a Piecewise Hölder Refractive Index in a Special Function Class

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The problem of reconstructing a piecewise Hölder continuous function describing the refractive index of an inhomogeneous obstacle scattering a monochromatic wave is considered. The boundary value scattering problem is reduced to a system of integral equations. The equivalence of the integral and differential formulations of the problem is proved. A two-step method for solving the inverse problem is proposed. A linear integral equation of the first kind is solved at the first step. Sufficient conditions for the uniqueness of its solution in the class of piecewise constant functions are obtained. At the second step, the unknown refractive index is explicitly expressed in terms of the solution obtained at the first step.