Transition Layer for the Heterogeneous Allen-Cahn Equation

Abstract

We consider the equation (1) ε∆u = (u− a(x))(u − 1) in Ω, ∂u ∂ν = 0 on ∂Ω, where Ω is a smooth and bounded domain in R, ν the outer unit normal to ∂Ω, and a a smooth function satisfying −1 < a(x) < 1 in Ω. We set K, Ω+ and Ω− to be respectively the zero-level set of a, {a > 0} and {a < 0}. Assuming ∇a 6= 0 on K and a 6= 0 on ∂Ω, we show that there exists a… (More)

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