Clustering layers for the Fife—Greenlee problem in ℝⁿ

Abstract

We consider the following Fife–Greenlee problem: where Ω is a smooth and bounded domain in ℝn, ν is the outer unit normal to ∂Ω and a is a smooth function satisfying a(x) ∈ (–1, 1) in . Let K, Ω– and Ω+ be the zero-level sets of a, {a < 0} and {a < 0}, respectively. We assume ∇a ≠ 0 on K. Fife and Greenlee constructed stable layer solutions, while del Pino et al. proved the existence of one unstable layer solution provided that some gap condition is satisfied. In this paper, for each odd integer m ≥ 3, we prove the existence of a sequence ε = εj → 0, and a solution with m-transition layers near K. The distance of any two layers is O(ε log(1/ε)). Furthermore, converges uniformly to ±1 on the compact sets of Ω± as j → +∞

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

“…The layer with the asymptotics in (1.6) in this scalar problem is meaningful in describing pattern-formation for reactiondiffusion systems such as Gierer-Meinhardt with saturation, see [12,19,32,37] and the references therein. Recently this problem has been completely solved by del Pino-Kowalczyk-Wei [14] (in the two dimensional domain case) and Mahmoudi-Malchiodi-Wei [27] (in the higher dimensional case), see also [18].…”

“…18) and the cut-off function χ 0 is defined by χ 0 (η) = 1 if |η| < 1/8 and χ 0 (η) = 0 if |η| ≥ 1/4.Note that φ 2 is an odd function in the variable x for each z ∈ (0, 1/ ).…”

Locations of interior transition layers to inhomogeneous transition problems in higher -dimensional domains

“…For the corresponding fractional Laplacian, layer solutions are constructed in [14]. For and given by (1.3), there are many known existence results of transition layer solutions, see [2–4, 6–12, 17, 18, 22, 31].…”

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,…”