We address a boundary-value problem involving a Poisson–Boltzmann equation that models the electrostatic potential of a channel formed by parallel plates with an electrolyte solution confined between the plates. We show the existence and uniqueness of solution to the problem, with special (particular) solutions as bounds, namely, a Debye–Hückel type solution as lower bound and a Gouy–Chapman type solution as upper bound. Our results are based on the maximum principle for elliptic equations and are useful for characterizing the behavior of the solutions. Also, we introduce a numerical scheme based on the Chebyshev pseudo-spectral method to calculate approximate solutions. This method is applied in conjunction with a multidomain procedure that attempts to capture the dramatic exponential increase/decay of the solution near the plates.

我们解决了涉及Poisson-Boltzmann方程的边界值问题，该方程可对平行板形成的通道的静电势进行建模，而平行板的电解质溶液则被限制在板之间。我们以问题的解决方案（特定的）作为边界，即以Debye-Hückel型解决方案为下界，以Gouy-Chapman型解决方案为上界，证明了该问题解决方案的存在性和唯一性。我们的结果基于椭圆方程的最大原理，对于表征解的行为很有用。此外，我们介绍了一种基于Chebyshev伪谱方法的数值方案来计算近似解。该方法与尝试捕获板附近溶液的急剧指数增加/衰减的多域过程结合使用。