The paper is devoted to the existence and rigorous homogenisation of the generalised Poisson–Nernst–Planck problem describing the transport of charged species in a two-phase domain. By this, inhomogeneous conditions are supposed at the interface between the pore and solid phases. The solution of the doubly non-linear cross-diffusion model is discontinuous and allows a jump across the phase interface. To prove an averaged problem, the two-scale convergence method over periodic cells is applied and formulated simultaneously in the two phases and at the interface. In the limit, we obtain a non-linear system of equations with averaged matrices of the coefficients, which are based on cell problems due to diffusivity, permittivity and interface electric flux. The first-order corrector due to the inhomogeneous interface condition is derived as the solution to a non-local problem.

中文翻译：

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具有非线性界面条件的广义泊松-能斯特-普朗克问题的存在性和两尺度收敛性

本文致力于描述广义 Poisson-Nernst-Planck 问题的存在和严格同质化，该问题描述了两相域中带电物质的传输。由此，在孔相和固相之间的界面处假定条件不均匀。双非线性交叉扩散模型的解是不连续的并且允许跨越相界面的跳跃。为了证明平均问题，在两个阶段和界面处同时应用和制定了周期性单元上的两尺度收敛方法。在极限情况下，我们获得了一个非线性方程组，该方程组具有系数的平均矩阵，这些方程基于扩散率、介电常数和界面电通量引起的电池问题。