The transport of ions in an electrolyte solution creates the electrical double layer (EDL) which is formed at the charged surface. Such a structure having nanometer-scale thickness is essentially like a capacitor and plays an important role in many physical and electrochemical fields, such as semiconductors, electro-kinetic fluids and colloidal systems.

In order to investigate the behavior of the EDL structure, we treat a charge conserving Poisson-Boltzmann (CCPB) equation as the basis for a framework of the ion transport in electrolyte solutions and study its boundary layer solutions. The CCPB equation is a Poisson-Boltzmann type equation with nonlocal nonlinear coefficients derived from the steady-state Poisson-Nernst-Planck (PNP) equation under the conservation of ion concentrations. 

In this first lecture, we will introduce Existence, Uniqueness and Maximum principle of the CCPB equation with the Robin boundary condition in high-dimensional bounded domains.