The Research On Direct And Inverse Scattering Problems For Complex Scatterers

The dissertation mainly discusses direct and inverse time-harmonic acoustic scattering problems for complex scatterers. Firstly, we consider direct scattering problem from an impenetrable scatterer and a crack and the same problem from a penetrable scatterer and a crack where the scatterer D is a bounded domain in R2, the open arc Γ denotes a crack outside of D,aD E C2is the boundary of D, the wavenumbers k, k0and impendance cofficients κ,κ0,μ are positive,-k2is not Dirichlet eigenvalue of the Laplace operator in D and u satisfies the Sommerfeld radiation condition uniformly in x=x/|x|.We obtain the existence and uniqueness of the solutions to the above problems in three stages. The first step is to establish uniqueness of the problems; The second step is to reformulate the problems as a boundary integral system by using single-and double-layer potentials; In the third step, we show the main results on the existence and uniqueness of the solutions to the boundary integral system in accordance with the potential theory and Fredholm theory. Secondly, we take into account the direct and inverse scattering problem for a partially coated scatterer where D(?)R2is a bounded scatterer,Γ=ΓD∪ΓN∪ΓI∪S denotes the boundary of D, S(?)Γ is an isolated point set. After utilizing the above methods to obtain the existence and uniqueness of the solutions to the direct problem, we are focused on the mathematical basis of the inverse problem, i.e., we prove theoretically how to reconstruct the shape of the scatterer D by using the linear sampling method. The specific scheme is as follows. We set up a far field equation Fg=(?)∞at first, then decompose the far field operator into F=-FM-1H, at last prove that H has dense range and F is injective and has dense range.