The Question of Uniqueness for the Exterior Problems

Abstract

This chapter is concerned with two questions regarding the solutions of the homogeneous system (4.1). First, we state various results established in [24] on the radiation conditions imposed on certain solutions of (4.1) to guarantee the unique solvability of a number of boundary value problems associated with the system. In Theorem 4.4 we show that the single-layer and double-layer potentials introduced in Chapter 2 satisfy the radiation conditions, which means that seeking solutions in terms of these functions does not lead to problems of nonuniqueness. Second, we derive representation formulas for the solutions of (4.1) in both the interior and exterior domains. The interior representation formula (Theorem 4.5) is obtained straightforwardly. The exterior representation formula (Theorem 4.7) requires substantially more work; it is based on far-field estimates for the matrix of fundamental solutions that are deduced in Theorem 4.6 from the fact that each column of the matrix satisfies the radiation conditions.