Existence and uniqueness of steady-state solutions for an electrochemistry model (Q2701579)

The authors study existence and uniqueness of steady-state solutions for an electrochemistry model with multiple species. The species concentrations can be expressed in terms of the electrical potential \(\varphi\) and the problem is equivalent to solving an integro-differential equation for \(\varphi\).NEWLINENEWLINENEWLINEIf \(\Omega\) is an open and bounded \(C^{0,1}\) domain in \(R^n(1\leq n\leq 3)\) and \(\partial\Omega =\Gamma_1\cup \Gamma_2\cup \Gamma_3\) where \(\Gamma_1\) and \(\Gamma_2\) are relatively closed, then the equation for the potential \(\varphi\) is NEWLINE\[NEWLINE-\varepsilon\Delta \varphi= \sum^m_{i=1} {z_iC_ie^{-z_i \varphi(x)}\over \int_\Omega e^{-z_i \varphi(y)} dy}+Q(x) \tag{1}NEWLINE\]NEWLINE with boundary conditions NEWLINE\[NEWLINE\varphi= -{\alpha\over 2}\text{ on }\Gamma_1, \quad\varphi= {\alpha \over 2}\text{ on }\Gamma_2,\;{\partial\varphi \over\partial \nu}=0\text{ on } \Gamma_3 \text{ and }Q\in L^2(\Omega). \tag{2}NEWLINE\]NEWLINE The authors prove the existence using Schauder's fixed point theorem.