Minimization of Gibbs Free Enrgy for a Poisson-Boltzmann Problem

In this paper, we consider an ionic solution containing several types of spherical ions between two uniformly charged parallel planes. The concentration of the ions at electrostatic and thermal equilibrium is determined by minimizing the Gibbs free energy of the system. The minimization problem is infinite-dimensional and so it is discretized to a finite-dimensional one, which is then solved using a standard optimization solver. A convergence analysis of the discrete minimum to the continuous minimum is performed.