The Mixed Scattering Problem Of A Crack And An Obstacle With An Impenetrable Core

In this paper,we study a class of mixed scattering problems of acoustic waves in R².The scattering obstacle consists of two parts,one of which is a smooth and bounded obstacle D₁,containing an impenetrable core D₀,and the other part is a crack ? which is outside of the obstacle D₁.The direct problem can be described as:We suppose f₁?H^1/2((?)Di),f₂? H^-1/2((?)D₁),f₃?H^1/2(?),f₄?H^-1/2(?),our work is to find functions u(x)?H_loc¹(R²\(D₁??)),u(x)?H₁(D₁\(?)₀)meeting those conditions(?)where r=|x|,and the convergence holds uniformly in x=x/|x|.We discuss this mixed scattering problem in two parts.We prove the well-posedness of the solution to the direct problem,and establish the theoretical basis of linear sampling method to study the inverse problem,Correspondingly,the numerical simulation is carried out by MATLAB combined with practical examples.When we consider the direct problem,the uniqueness of the solution to the above question can be proved by using Rellich lemma and Green's formula firstly.Then the general for-mal solution to this problem is constructed by potential theory,we are able to prove the existence of the solution by the boundary integral equation method.Moreover,in the study of the inverse problem,Linear sampling method is used to reconstruct the shape of the crack ? and obstacle D₁.