Green’s functions representation

Existence and uniqueness of solutions to the b.v.p. analogous to (2.2.1) has been proved in numerous contexts (see, e.g., [2]—[6]) and can be easily inferred for (2.2.1). We shall not do it here. Instead we shall assume the existence and uniqueness for (2.2.1) and similar formulations and, based on this assumption, we shall discuss some simple properties of the appropriate solutions. These properties will follow from those of the solution of the one-dimensional Poisson-Boltzmann equation, combined with two elementary comparison theorems for the nonlinear Poisson equation. These theorems follow from the Green s function representation for the solution of the nonlinear Poisson equation with a monotonic right-hand side (or from the maximum principle arguments [20]) and may be stated as follows. 

The quantity uf r,a>yt) at any point is given by a Green s function representation, which we shall discuss in Sect. 2.3, involving integrals over the boundary functions at a given time. The time variable comes in only through these boundary functions. Let us write (2.1.8) in the form... 

These results will be referred to as the Generalized Partial Correspondence Principle, to distinguish it from the more detailed, and specialized. Extended Correspondence Principle. A more general derivation of this result, which does not rely on the Green s function representation of the solution, has been given by Graham and Golden (1988). 

We have intentionally omitted the details in progressing from Eq. (26-1) to Eq. (26-2) because of their algebraic complexity. Since we only want to show, the formal procedure for decomposing the radiation field, such details are counter-productive from a pedagogical point of view, and are to be found elsewhere [1-3]. Similarly, the approach from the Green s function representation to integrals of the form of Eq. (26-2) is available in Refs. [3,4]. 

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