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]).

This research was supported in part by grants from the Deutsche Forschungsgemeinschaft and the National Science Foundation.