Interfaces with boundary intersection for an inhomogeneous Allen-Cahn equation in three-dimensional case

Abstract

The following boundary value problem is considered: \begin{document}$ \begin{equation*} \left\{\begin{array}{ll} \varepsilon^2\Delta u+V(y)u(1-u^2) = 0, &\mbox{in}~~ \Omega , \\ \frac{\partial u}{\partial n} = 0, &\mbox{on }~~ \partial \Omega, \end{array}\right. \end{equation*} $\end{document} where $ \Omega\subset \mathbb R^3 $ is a smooth bounded domain. Assume that $ \Gamma\subset \bar{\Omega} $ is a smooth surface which intersects $ \partial\Omega $ at a right angle and separates $ \Omega $ into two parts $ \Omega_1 $ and $ \Omega_2 $. In addition, we also assume that $ \Gamma $ is a non-degenerate critical point of the functional $ \mathcal{K}(\Gamma) = \int_\Gamma V^{\frac12}d\mu $. From the infinite dimensional reduction method, we find a particular kind of solutions which converge to $ 1 $ in $ \Omega_1 $ and to $ -1 $ in $ \Omega_2 $ as $ \varepsilon\to 0 $.