On The Direct And Inverse Scattering Problems For Complex Scatterers Containing Cracks

Acoustic and electromagnetic waves are widely used in medical imaging, geophysical exploration, nondectructive testing, radar and other fields. Given the time harmonic incident wave, the direct problem is to study the well-posedness of scattering problem generated by the scatterer and verifying Helmholtz or Maxwell equations with certain boundary conditions. The inverse problem is often based on far-field information or other measurement data to reconstruct the location and shape of the scatterer with 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 cracks due to harmonic incident waves as well as the direct electromagnetic scattering for multi-layered background medium.The first chapter mainly introduces the basic overview of scattering theory, the basic problems and its conclusions during the development of scattering theory. The first sec-tion introduces the background and significance of scattering theory. The second section briefly introduces inverse problem. The third section describes the investigation situation-s of the direct scattering problem, including electromagnetic and acoustic mathematical models; the solving methods for direct problem, the uniqueness and the property of the far-field operators; and the research status of the inverse scattering problem, mainly on the well-posdeness, including the ill-posedness, existence, uniqueness, stability and algo-rithms. Section Ⅳ introduces the work of this thesis and structural arrangements, we mainly focus on the scattering problem which have a multi-layered structure containing cracks and mixed type scatterer.The second chapter introduces the basic tools, theorems and methods used in the scattering theory. The first section describes the special functions, incident waves, com-mon function spaces. Section Ⅱ introduces potential theory. Section Ⅲ describes the common basic theorems in scattering theory, including the Green representation theo-rem, Rellich lemma, Holmgren lemma, the unique continuation principle, Lax-Milgram theorem and Riesz-Fredholm theorem. Section Ⅳ presents the Tikhonov regularization method for the first class of integral equations. Section Ⅴ introduces the linear sampling method and outline its reconstruction steps under taking the inversion of impenetrable obstacle scattering problem for example.The third chapter studies the direct and inverse scattering problems by a crack with piecewise background. The research on direct scattering problem based on [126], using the boundary integral equation method, and employing Fredholm theorem, we prove the existence and uniqueness of the solution. The research on inverse scattering problem based on [97], [98] and [115], we firstly prove the crack and layer can be uniquely identified, and then use the linear sampling method to reconstruct the crack.The fourth chapter researches the direct and inverse scattering problem by a mixture scatterer which contains a crack. About the research on direct scattering problem, see Yan [142] and Yan and Yao [143] [144]. We firstly consider the scattering problem for a mixed scatterer composed by a crack and an impenetrable obstacle. Using variational method we establish the well-posedness of the direct problem, then we give an uniqueness result for the inverse problem and employ the linear sampling method to reconstruct the scatterer. Secondly, this chapter also considers the direct and inverse scattering problem of mixed scatterer which composed of a crack and an inhomogeneous media.Chapter V studies the direct electromagnetic scattering problem for multi-layered medium. We consider the scattering of time-harmonic electromagnetic plane waves by an impenetrable obstacle which is embedded in a piecewise homogeneous medium. Applying potential theory, the problem can be reformulated as a boundary integral system. We obtain the existence and uniqueness of the solution to the system by using the generalized Garding inequality and the Fredholm theory.