The Problem Of Scattering By A Mixture Of Obstacles And A Crack

This paper mainly studies the scattering problem containing a mixture of obsta-cles and a crack in R2.The problem can be described as:We suppose D1 is a bounded and penetrable area,D2 is a bounded and im-penetrable area in R2,Γ is a crack and for simplicity,we suppose it’s a paart of a closed curve(?)D3,and(?)Di(?)C2(i=1,2,3)are all smooth curves.We give certain boundary conditions on(?)D1,(?)D2,Γ,then we can get a mixed boundary problem about Helmholtz equation.We suppose f1∈H1/2((?)D1),f2 ∈H1/2((?)D2),f3∈H1/2(Γ),the problem is to find a solution u ∈ H1(D1)∪Hloc1(R2\(D1∪D2∪Γ))meeting the following question:(?)and u satisfy Sommerfeld radiation condition at infinity:uniformly for all directions x = x/|x|.We mainly study this problem in three steps:The first step:prove the uniqueness of the solution;The second step:by using the boundary integral equation approach and potential theory,we transform the problem into a equivalent boundary integral system and then prove the existence of the problem;The third step:by using the boundary integral equation approach and linear sampling method,we study the corresponding inverse problem.By using the far-field information u∞,we can reconstruct the shape of obstacles D1,D2 and the crackΓ,then we give the numerical implementation of some simple examples.