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.

逆障礙散射問題包括從散射數據中找到一個物體（障礙物）的未知表面，其中散射振幅（(;;)）是指散射振幅，; 是指散射和入射波的方向， 是指ℝ3中的單位球，而k > 0是波矢量的模數。

如果散射數據的維度與未知物體的維度相同，則稱其為非過度確定。維度是指描述數據或物體的函數的最小變量數量。在逆障礙散射問題中，這個數量是2，非過度確定數據的一個例子是(): = (;₀;₀)。通過下標0表示變量的固定值。

本書證明了對於在的開放子集中已知的數據()，可以唯一確定表面和上的邊界條件。這個條件可以是迪利克雷條件、諾伊曼條件或阻抗類型。

上述的唯一性定理非常重要，因為非過度確定數據是唯一確定未知的最小數據。文獻中沒有這樣的結果，因此需要這本書。本書包含了粗糙表面散射解存在性和唯一性的自包含證明。