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For the structure of the thin electrical double layer~(EDL) and the propertyrelated to the EDL capacitance, we analyze boundary layer solutions(corresponding to the electrostatic potential) of a non-local elliptic equationwhich is a steady-state Poisson--Nernst--Planck equation with a singularperturbation parameter related to the small Debye screening length.Theoretically, the boundary layer solutions describe that those ions exactlyapproach neutrality in the bulk, and the extra charges are accumulated near thecharged surface. Hence, the non-neutral phenomenon merely occurs near thecharged surface. To investigate such phenomena, we develop new analysistechniques to investigate thin boundary layer structures. A series of fineestimates combining the Poho\v{z}aev's identity, the inverse H\"{o}lder typeestimates and some technical comparison arguments are developed in arbitrarybounded domains. Moreover, we focus on the physical domain being a ball withthe simplest geometry and gain a clear picture on the effect of the curvatureon the boundary layer solutions. In particular, for the cylindrical electrode,our result has a same analogous measurement as the specific capacitance of thewell-known Helmholtz double layer.

Thin layer analysis of a non-local model for the double layer structure