ABSTRACT
The expression −ih′T + iH ′T appears as the normal derivative of h+H in the proof of Theorem 18.1. In a simply connected domain, we will see that the condition −ih′T + iH ′T = 0 forces h′ ≡ 0 and H ′ ≡ 0. Before we treat the Neumann problem in a multiply connected domain, we must determine which functions h and H in H2(bΩ) satisfy the condition −ihT + iHT = 0, i.e., satisfy hT = HT . By Theorem 4.3, a function u that is expressible as both hT and HT would be orthogonal to H2(bΩ) and orthogonal to conjugates of functions in H2(bΩ). We will see that such functions are closely related to the classical harmonic measure functions (which we define below).
