We consider the nonlinear problem of inhomogeneous Allen-Cahn equation $ \epsilon^2\Delta u+V(y)\,(1-u^2)\,u = 0\quad \mbox{in}\ \Omega, \qquad \frac {\partial u}{\partial \nu} = 0\quad \mbox{on}\ \partial \Omega, $ where $ \Omega $ is a bounded domain in $ \mathbb R^2 $ with smooth boundary, $ \epsilon $ is a small positive parameter, $ \nu $ denotes the unit outward normal of $ \partial \Omega $, $ V $ is a positive smooth function on $ \bar\Omega $. Let $ \Gamma\subset\Omega $ be a smooth curve dividing $ \Omega $ into two disjoint regions and intersecting orthogonally with $ \partial\Omega...
