Asymptotic Analysis of Boundary Layer Solutions to Poisson-Boltzmann Type Equations in General Bounded Smooth Domains

Abstract

We study the boundary layer solution to singular perturbation problems involving Poisson-Boltzmann (PB) type equations with a small parameter $\epsilon$ in general bounded smooth domains (including multiply connected domains) under the Robin boundary condition. The PB type equations include the classical PB, modified PB and charge-conserving PB (CCPB) equations, which are mathematical models for the electric potential and ion distributions. The CCPB equations present particular analytical challenges due to their nonlocal nonlinearity introduced through integral terms enforcing charge conservation. Using the principal coordinate system, exponential-type estimates and the moving plane agruments, we rigorously prove asymptotic expansions of boundary layer solutions throughout the whole domain. The solution domain is partitioned into three characteristic regions based on the distance from the boundary: Region I, where the distance from the boundary is at most $T\sqrt\epsilon$, Region II, where the distance ranges between $T\sqrt\epsilon$ and $\epsilon^\beta$, and Region III, where the distance is at least $\epsilon^\beta$, for given parameters $T>0$ and $0<\beta<1/2$. In Region I, we derive second-order asymptotic formulas explicitly incorporating the effects of boundary mean curvature, while exponential decay estimates are established for Regions II and III. Furthermore, we obtain asymptotic expansions for key physical quantities, including the electric potential, electric field, total ionic charge density and total ionic charge, revealing how domain geometry regulates electrostatic interactions.