Boundary integral equations for mixed Dirichlet, Neumann and transmission problems

We consider a Helmholtz equation in a number of Lipschitz domains in n ≥ 2 dimensions, on the boundaries of which Dirichlet, Neumann and transmission conditions are imposed. For this problem an equivalent system of boundary integral equations is derived which directly yields the Cauchy data of the solutions. The operator of this system is proved to be injective and strongly elliptic, hence it is also bijective and the original problem has a unique solution. For two examples (a mixed Dirichlet and transmission problem and the transmission problem for four quadrants in the plane) the boundary integral operators and the treatment of the compatibility conditions are described.