Modified Green's functions and the third boundary value problem for the Helmholtz equation

Abstract

It is well known that the application of boundary integral equation methods to the exterior Dirichlet and Neumann boundary value problems for the Helmholtz equation results in Fredholm equations which fail to have unique solutions at eigenvalues of the interior adjoint problem. As has often been pointed out, the non-uniqueness is an artifact of the integral equation method since the exterior boundary value problems are themselves uniquely solvable. A number of devices have been proposed to produce solutions to these exterior boundary value problems using integral equations which do have unique solutions (see, for example, Brakhage and Werner [ 11. Burton and Miller 121, Jones [3], Kleinman and Roach [4], Leis 151, and Ursell 161). Recently Kleinman and Roach [7,8] have discussed modified Green’s functions for the Dirichlet and Neumann problems and have shown how, by appropriate choice of parameter, one may not only produce a modification of the kernel which is free from real eigenvalues but also is a best possible choice according to one of a number of specified criteria, e.g., closest in the L, operator norm to the true Green’s function. In this note we extend this analysis to the more complicated case of the Robin or third boundary value problem. In doing so we follow the analysis of our earlier paper [9] in which we treated this boundary value problem for generalized L, boundary data. We begin with a more precise description of the problem and a summary of the results of this latter paper which we will find useful.