For the structure of the thin electrical double layer (EDL) and the property related to the EDL capacitance, we analyze boundary layer solutions (for the electrostatic potential) of a charge-conserving Poisson-Boltzmann (CCPB) equation which is a steady-state Poisson-Nernst-Planck equation with a singular perturbation parameter related to the small Debye screening length. Theoretically, the boundary layer solutions describe that those ions exactly approach neutrality in the bulk, and the extra charges are accumulated near the charged surface. Hence, the non-neutral phenomenon merely occurs near the charged surface. In this talk, we introduce new analysis techniques to investigate thin boundary layer structures. A series of fine estimates combining the Pohozaev's identity, the inverse Holder type estimates and some technical comparison arguments are developed in arbitrary bounded domains. We further concentrate on the physical domain being a ball with the simplest geometry and gain a clear picture on the effect of the curvature on the boundary layer solutions. As an application, we provide a theoretical way to support that the EDL has higher capacitance in a quite thin region near the charged surface, not in the whole EDL. In particular, for the cylindrical electrode, our result has a same analogous measurement as the specific capacitance of the well-known Helmholtz double layer.

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