The Factorization Method Applied To Complex Scatterer Containing Penetrable Medium

When the time harmonic incident wave scattered by scatterer,the total field satisfies the Helmholtz equation or Maxwell equation and some boundary conditions due to the properties of the scatterer,so the direct scattering problem is a boundary value problem of the Helmholtz equation or Maxwell equation.And the inverse scattering problem is often based on far-field information or other measurement data to reconstruct the location and shape of the scatterer and its physical properties.However,the scatterer often has a complex structure,in this thesis,we will study the direct and inverse scattering problem for complex scatterer containing penetrable medium.Firstly,we introduce the basic overview of scattering theory,the basic problems and its conclusions during the development of scattering theory,including the background and significance of the inverse scattering theory,the mathematical models for scattering of acoustic and electromagnetic waves,the investigation situations and the solving methods of the direct and inverse scattering problem,the basic tools,theorems and methods used in the scattering theory,the basic ideas and principles of the factorization method and the main work of this thesis.Next,we introduce our work in three aspect in detail.In the first,we study the direct and inverse scattering problems by a mixed scatterer which contains a penetrable medium and an impenetrable obstacle.Firstly,the well-posedness of the direct problem is verified by using the boundary integral equation method and Fredholm theorem.Secondly,we derive the decomposition of far field operator F which satisfies the demand of Theorem 1.7.2,hence we can reconstruct the penetrable medium and the impenetrable obstacle simultaneously by the factorization method.In the last,we present some numerical examples to demonstrate the feasibility and effectiveness of our approach.In the second,we research the direct and inverse scattering problems by a mixed scatterer which contains a penetrable medium and an open arc.We show the existence and uniqueness of the solution to the direct problem by the Fredholm theorem and Rellich theorem respectively.The inverse scattering problem is influenced by ? which is the mass density ratio of the media in the exterior of the mixed scatterer and the penetrable medium.When 0<?<1,we derive suitable decomposition of the far field operator F;when ?>1,we decompose the auxiliary operators Fm and Fl.Finally,we reconstruct the shape of the penetrable medium and the open arc and present some numerical examples.In the third,we consider a kind of exterior transmission problem in which the re-fractive index n(x)is a piecewise positive constant.Through establishing an equivalent boundary integral system,we obtain that the set of exterior transmission eigenvalues is a discrete set.Furthermore,we prove that there does not exist a transmission eigenvalue under some conditions.