Inverse Obstacle Scattering with Modulus of the Far Field Pattern as Data

Abstract

For the two-dimensional inverse scattering problem for a sound-soft or perfectly conducting obstacle we may distinguish between uniqueness results on three different levels. Consider the scattering of a plane wave u i(x) = e ikx·d (with wave number k > 0 and direction d of propagation) by an obstacle D, that is, a bounded domain D ⊂ IR2 with a connected boundary ∂D. Then the total wave u is given by the superposition u = u i + u s of the incident wave u i and the scattered wave u s and obtained through the solution of the Helmholtz equation $$\bigtriangleup u+k^{2} u = \textup{0 in IR}^{2} \setminus \bar{D}$$ subject to the Dirichlet boundary condition $$u = \textup{0 on }\partial D$$ and the Sommerfeld radiation condition $$\underset{r\rightarrow \infty} {\textup{lim}}\sqrt{r}\left ( \frac{\partial u^{s}}{\partial r}-iku^{s} \right )= 0, r = \left | x \right |$$ , uniformly with respect to all directions. The exterior Dirichlet problem (1)–(3) has a unique solution provided the boundary ∂D is of class C 2 (see [1, 2]).