A boundary integral equation procedure for the mixed boundary value problem of the vector helmholtz equation

A boundary integral method is developed for the mixed boundary value problem for the vector Helmholtz equation in R³. The obtained boundary integral equations for the unknown Cauchy data build a strong elliptic system of pseudodifferential equations which can therefore be used for numerical computations using Galerkin's procedure. We show existence, uniqueness and regularity of the solution of the integral equations. Especially we give the local "edge" behavior of the solution near the submanifold which divides the Dirichlet boundary from the Neumann boundary