THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN

We consider a kind of scattering problem by a crack Γ that is buried in a bounded domain D,and we put a point source inside the domain D.This leads to a mixed boundary value problem to the Helmholtz equation in the domain D with a crack Γ.Both sides of the crack Γ are given Dirichlet-impedance boundary conditions,and different boundary condition(Dirichlet,Neumann or Impedance boundary condition) is set on the boundary of D.Applying potential theory,the problem can be reformulated as a system of boundary integral equations.We establish the existence and uniqueness of the solution to the system by using the Fredholm theory.