Phase Transition Layers For Fife-greenlee Problem

We consider the Fife-Greenlee problemε2 △u + (u - a(y)) (1 - u2) = 0 in α,(?)u/(?)v=0 on (?)α,where Ω is a bounded domain in R2 with smooth boundary, ε > 0 is a small positive parameter, v denotes the unit outward normal of (?)Ω. Let Γ {y∈Ω :a(y) = 0}be a simple smooth curve intersecting orthogonally with (?)Ω at exactly two points and dividing Ω into two parts. We assume that - 1 < a(y) < 1 on Ω and ▽a ≠0 on Γ and also some admissibility conditions between Γ and (?)Ω. We can prove that there exists a solution uε such that: as ε → 0,uε approaches +1 in one part, while tends to -1 in the other part, except a small neighborhood of Γ.