On The Direct And Inverse Scattering Problem For Cracks In Mixed Scatterer

The direct and inverse scattering problem of time-harmonic acoustic or electromagnet-ic waves is a famous mathematical physics problem.This theory is widely used in medical imaging,nondestructive test,geological exploration,reconnaissance radar,remote sens-ing and environmental monitoring fields.Simply speaking,the direct scattering problem is to determine the scattered waves for given incident wave and scatterer,whose well-posedness governed by Helmholtz equation or Maxwell equations with certain boundary condition;while the inverse scattering problem is to reconstruct the location,shape and physical properties of the scatterer from the measured data(far field or near field).This paper mainly studies the direct and inverse scattering problem for three kinds of mixed scatterer containing cracks as following:· Mixed scattering problem for cracks and penetrable inhomogeneous medi-um.Consider the scattering problem of time-harmonic plane waves by a union of a crack and a penetrable inhomogeneous medium with compact support.The well-posedness of the direct problem is established by the variational method.An u-niquencss result for the inverse problem is proved,that is,both the crack and the inhomogeneous medium can be uniquely determined by a knowledge of the far field pattern for incident plane waves.The linear sampling method is employed to recover the location and shape of the combined scatterer.It is worth noting that we make the first step on reconstructing a mixed-type scatterer of a crack and an inhomogeneous medium by the linear sampling method.· Mixed scattering problem for cracks surrounded by inhomogeneous medi-um.Consider the inverse scattering problem of determining the shape and location of a crack surrounded by an known inhomogeneous medium.Both the Dirichlet boundary condition and a mixed type boundary condition are considered.In order to avoid using the background Green function in the inversion process,a reciprocity relationship between the Green's function and the solution of an auxiliary scattering problem is proved.Then we focus on extending the factorization method to our inverse shape reconstruction problems by using far field measurements at fixed wave number.· Mixed scattering problem for multiple cracks.Consider the scattering problem of timc-harmonic plane waves by two disjoint cracks with mixed boundary condition.Having established the well-posedness of the solution to the direct problem by inte-gral equation method,we prove an uniqueness result for the inverse problem,and reconstruct two cracks and its parameter simultaneously by linear sampling method.In addition,we extend Hassen's work to the construction of curve's normals and give some numerical results for multiple cracks.