A traditional model (1910) is the Gouy-Chapman model, which predicts the distribution of ions and the shape of a poential in a solution, assumed to be semi-infinite. This model, refined by Stern in 1924, takes into account the Debye length. We propose a variational formulation of the Poisson-Boltzmann system, both for electroneutral and non electroneutral cells. We prove the existence and uniqueness of a solution in the 1d setup (radial solutions in 2d and 3d are included), give ideas for the existence and uniqueness of a solution in higher dimension, and obtain an analytic solution in the 1d set-up for non electroneutral cells. This analytic solution relies on incomplete Jacobi elliptic functions, and the unique solution is obtained by solving a system of equations in $\R_+^2$ (two equations with two unknowns)

Joint work with Clair Poignar (Inria Bordeaux Sud Ouest)

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