The Problem Of Mixed Acoustic Wave Scattering By Penetrable Media With An Impenetrable Core And A Crack

In this paper,we mainly study the scattering of mixed acoustic waves by pen?etrating media with an impenetrable core and a crack.We suppose D2 is an open and bounded domain in R2 with smooth boundary,it contains an impenetrable core D1 that is D1(?)D2,? is a bounded crack in R2.D2?? are disjoint.The direct scattering problem is described as follows.We suppose f1?H1/2((?)D2),f2 ? H-1/2((?)D2),f3?H1/2(F),f4?H-1/2(?),we need to seek u(x)? Hloc1(R2\(D2 ? ?)),w(x)?H1(D2\D1)satisfying following problem:(?)where r=|x|,and the convergence condition holds uniformly in x=x/|x|.The main content of this paper includes two aspects:one is the well-posedness of the solution of the positive problem,that is,the existence and uniqueness of the solution of the above problem.In the proof of uniqueness,Green's formula and Relich's lemma are used,and the method of boundary integral equation is used to prove the existence.The other content is the inverse scattering problem,which is to reconstruct the shape of D2 and ? through far-field information.We use linear sampling method.