Path Argument for Inverse Acoustic and Electromagnetic Obstacle Scattering Problems

Abstract

Let us recall the acoustic obstacle scattering problem from sound-hard or sound-soft obstacles. To describe the scattering system, we shall use uⁱ, u^s and u to represent the incident, scattered and total field, respectively, where u = uⁱ + u^s and $u^i(x)=\exp \{\mathrm ik\mathbf { x}\cdot \mathbf { d}\}$ with $\mathrm i=\sqrt {-1},\ \mathbf { d}\in \mathbb S^{N-1}$ being the incident direction and k > 0 being the wave number. Let D be an impenetrable obstacle, where D is a general compact set in $\mathbb R^N(N\geq 2)$ with an open connect complement $G=\mathbb R^N\setminus D$.