MINIMIZATION OF FREE ENERGY IN A POISSON–BOLTZMANN PROBLEM

In this paper, we study a mathematical model in which an ionic solution composed of several types of spherical ions is confined between two uniformly charged parallel planes. The ion concentrations at equilibrium, determined by the interplay between thermal and electrostatic effects, are obtained by minimizing the free energy of the system. The resulting minimization problem is infinite-dimensional, and we prove the existence and uniqueness of its solution by means of a norm-convergence direct method based on the Banach–Saks property of the space in which the problem is posed. The existence proof is technically challenging because the functional to be minimized is not coercive in general. Three illustrative problems are considered and solved numerically.