Asymptotic behavior of a solution to the Poisson--Boltzmann equation in a three-dimensional domain with a thin bridge (Q1271984)

The following mixed boundary value problem for the Poisson--Boltzmann equation is studied: \[ -\Delta_xu(x) + \lambda \big(\exp (\beta u(x)\big) - \exp \big(-\beta u(x)\big) = 0, \quad x\in\Omega_{\varepsilon}; \] \[ u(x) = 1, \quad x\in\Sigma_{\varepsilon};\qquad u(x) = 0, \quad x\in\Sigma;\quad \partial_nu(x) = 0,\quad x\in S_{\varepsilon}. \] Here \(\Omega_{\varepsilon} = \{x=(y,z)\in \mathbb{R}^3: y\in\omega,\;0<z<\varepsilon + h(y)\}\), \(\Sigma = \{x: y\in\omega, \;z=0\}\), \(\Sigma_{\varepsilon} = \{x: y\in\partial\omega ,0<z<\varepsilon + h(y)\}\), where \(\omega\subset \mathbb{R}^2\) is a bounded domain and \(S_{\varepsilon}\) denotes the lateral area of the quasicylinder \(\Omega_{\varepsilon}\). The aim of the article is to construct and justify an asymptotic expansion (as \(\varepsilon\to +0\)) for the solution of the above problem in a neighborhood about the origin. The problem involves two additional parameters that have strong influence on the construction of the asymptotic expansion. Several typical situations are discussed in detail.