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

Abstract

We consider the nonlinear problem of inhomogeneous Allen-Cahn equation 2 ∆u + V (y) (1 − u 2) u = 0 in Ω, ∂u ∂ν = 0 on ∂Ω,

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

“…The fact is due to the effect of ∂Ω toward the boundary layers caused by the homogeneous Neumann boundary condition in (1.1). This is quite different from the results in previous papers (such as [14], [15] on clustering interfaces for (1.3), and also [46], [47] on interior clustering interfaces for (1.1)) in which multiple layers in the cluster distribute evenly along the limit set (The distances between neighbouring layers are almost the same in the main order).…”

“…Fan, B. Xu and J. Yang [22] constructed a solution with single interior phase transition layer connecting ∂Ω near a curve Γ, which connects perpendicularly the boundary ∂Ω and is also stationary and non-degenerate with respect to Γ V 1 2 . Clustered interior phase transition layers connecting ∂Ω can be found in the paper by S. Wei and J. Yang [46].…”