Electric field and force saturation

This result, which in the one-dimensional case (for parallel plates) follows from a direct computation (e.g., [9]), is shown below (Proposition 2.1), for parallel cylinders and for spheres. 

The developments of 2.2.2 and 2.2.3 employ some elementary comparison results (Theorems 2.1, 2.2) of 2.2.2. These and much more general comparison results are well known for solutions of (2.1.3a) (e.g., [4]). They are presented in 2.2.2 for the sake of completeness. The same applies to the nonnegativeness part of Proposition 2.1, 2.2.2. 

Field saturation. Consider a particle occupying a convex open domain CR3 (or ft2) with a smooth boundary du , charged to the electric potential 0, at equilibrium with an infinite solution of a symmetric electrolyte of a given average concentration. (Properties described below are directly generalizable to an arbitrary electrolyte or electrolyte mixture.) 

The equilibrium electric potential ip in the space surrounding the particle is described by the following b.v.p. 

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. 

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