Boundary Integral Equations for the Three-Dimensional Helmholtz Equation

The relation between various boundary integral equation formulations of Dirichlet and Neumann problems for the three-dimensional Helmholtz equation is clarified. The integral equations derived using single or double layer distributions as well as those based on the Helmholtz representation using an unmodified free space Green’s function are presented in uniform notation; the connection between interior and exterior problems is demonstrated; the spectra of the associated integral operators are investigated; and various properties of interior eigenfunctions are derived. Assuming only that exterior problems have at most one solution, existence as well as uniqueness of the corresponding boundary integral equations is established even at eigenvalues of the adjoins interior problems. In this case, supplementary conditions are required, and these are presented in the form of boundary rather than volume integral equations.