Phase transition layers with boundary intersection for an inhomogeneous Allen–Cahn equation

Abstract

We consider the nonlinear problem of inhomogeneous Allen–Cahn equationϵ2Δu+V(y)u(1−u2)=0inΩ,∂u∂ν=0on∂Ω, where Ω is a bounded domain in R2 with smooth boundary, ϵ is a small positive parameter, ν denotes the unit outward normal of ∂Ω, V is a positive smooth function on Ω¯. Let Γ be a curve intersecting orthogonally with ∂Ω at exactly two points and dividing Ω into two parts. Moreover, Γ satisfies stationary and non-degenerate conditions with respect to the functional ∫ΓV1/2. We can prove that there exists a solution uϵ such that: as ϵ→0, uϵ approaches +1 in one part of Ω, while tends to −1 in the other part, except a small neighborhood of Γ.