Mixed Scattering Problem Of Penetrating Matter With Impenetrable Core

In this paper,we mainly study the scattering of mixed acoustic waves by a pen-etrable material with an impenetrable core and another impenetrable material.We suppose ?1 is an open and bounded domain in R2 with smooth boundary,it contains an impenetrable core ?0 that is ?(?)?1,and ?2 is an impenetrable obstacle with smooth boundary.And ?i ? ?2=(?).So,the direct scattering problem is described as follows.We suppose h0?H1/2((?)?1)?h1?H-1/2((?)?1)?h2?H-1/2((?)?2),we need to seek?(x)?Hloc1(R2\(?1??2))?v(x)?H1(?0)satisfying following problem:(?)where r=|x| and the convergence condition holds uniformly in x=x/|x|.For this problem,we mainly study the positive scattering problem and the inverse scattering problem.Concerning the positive scattering problem,we mainly prove the existence and uniqueness of the solution,in which the uniqueness proof mainly uses Green for-mula and Rellich lemma.The existence proof uses the boundary integral equation method,that is,using Green formula to construct the solution.Then,the poten-tial theory is used to transform the original problem into corresponding boundary integral equations,and the existence of solutions is proved.With regard to the inverse scattering problem,the far-field equation is estab-lished by processing the far-field information,and then discretized,and finally re-constructed by linear sampling method to reconstruct the shape of ?1 and ?2.