Abstract
In this work, we provide a proof of the so-called Karp's theorem in a different approach. We use the unique continuation principle together with the monotonicity of eigenvalues for the negative Laplace operator. This method is new and would be applicable to other types of inverse scattering problems.
Highlights
The inverse obstacle scattering problem deals with the determination of shapes and/or positions of objects from the knowledge of associated scattered fields, which has many applications in areas, such as medical imaging, seismic imaging, and non-destructive testing
We refer the reader to his series of work, for instance, see [8, 9, 10] and the references therein
By taking ε to be sufficiently small, we show u = 0 in Dε, which yields a contradiction
Summary
The inverse obstacle scattering problem deals with the determination of shapes and/or positions of objects from the knowledge of associated scattered fields, which has many applications in areas, such as medical imaging, seismic imaging, and non-destructive testing. The total field u, depending on the nature of the scatter, satisfies one of the following boundary conditions:. Due to the radiation condition (1b), one can define the far-field pattern u∞(x, d, k), which is a complex-valued function of receiver direction x = x/r, incident direction d, and wave number k > 0 as follows. The inverse obstacle scattering problem is formulated to determine ∂Ω from u∞(x, d, k) defined on a certain subset E of S1 × S1 × R1+. We refer the reader to his series of work, for instance, see [8, 9, 10] and the references therein The purpose of this short note is to provide a proof of Karp’s theorem using a different approach, to advance our understanding of obstacle scattering phenomena.
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