The Direct Scattering Problem For Two Scatterers With Different Boundary Conditions

This paper focuses on a class of mixed wave scattering problem, for the convenience of our study, we only discuss this issue in R2. Assumed that the scattering body is composed by two independent bounded scatterers, and they have different boundary conditions, specific problems can be described as follows:Assume that D and D1are two smooth bounded domains, this problem can be attributed to the outer and mixed boundary value problem of Helmholtz equation in R2. Namely:given To identify a u∈Hloc1(R2\(D∪D1))satisfy: And the scattered field also meet the conditions of Sommerfeld attenuationFrom this paper, we gained the existence and uniqueness of the solutions for the above problem. First, by using the Rellich Lemma, we can get the uniqueness of the solutions for the problems; Secondly, The focuses of this article is that by using boundary integral equation method to get the existence of the solutions for the problems. Then by the above two steps, the existence and uniqueness of solutions for the problems can be solved.