Phase transition layers for Fife-Greenlee problem on smooth bounded domain

Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation

“…Modified-Fermi-Coordinates system. In this section, we first recall the coordinate system in a neighborhood of Γ from [35], called modified Fermi coordinates, which was also used in [32] to construct interfaces for Fife-Greenlee problem on bounded smooth domain. The local coordinate system can be given as follows.…”

Clustering of Boundary Interfaces for an inhomogeneous Allen-Cahn equation on a smooth bounded domain

“…Problem (1.2) has been studied extensively in recent years. See [1,2,5,7,8,9,10,11,13,16,17,18,21,23,24,26,30,43,44] for backgrounds and references. The case V (y) ≡ 1 or a(y) = 0 corresponds to the Allen-Cahn equation [6] ǫ 2 ∆u + u(1 − u 2 ) = 0 in Ω, ∂u ∂ν = 0 on ∂Ω,…”