Inverse Obstacle Scattering with Non-Over-Determined Scattering Data

The inverse obstacle scattering problem consists of finding the unknown surface of a body (obstacle) from the scattering ��(��;��;��), where ��(��;��;��) is the scattering amplitude, ��;�� �� �� is the direction of the scattered, incident wave, respectively, �� is the unit sphere in the ℝ3 and k > 0 is the modulus of the wave vector.

The scattering data is called non-over-determined if its dimensionality is the same as the one of the unknown object. By the dimensionality one understands the minimal number of variables of a function describing the data or an object. In an inverse obstacle scattering problem this number is 2, and an example of non-over-determined data is ��(��): = ��(��;��₀;��₀). By sub-index 0 a fixed value of a variable is denoted.

It is proved in this book that the data ��(��), known for all �� in an open subset of �� , determines uniquely the surface �� and the boundary condition on ��. This condition can be the Dirichlet, or the Neumann, or the impedance type.

The above uniqueness theorem is of principal importance because the non-over-determined data are the minimal data determining uniquely the unknown ��. There were no such results in the literature, therefore the need for this book arose. This book contains a self-contained proof of the existence and uniqueness of the scattering solution for rough surfaces.