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

Feifei Tang; Suting Wei; Jun Yang

doi:10.3934/dcds.2018063

Article Contents

2018, Volume 38, Issue 3: 1527-1552. Doi: 10.3934/dcds.2018063

This issue Previous Article Existence of nonnegative solutions to singular elliptic problems, a variational approach Next Article On the universality of the incompressible Euler equation on compact manifolds

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

1.
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
2.
School of Mathematics and Statistics, & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

* Corresponding author: Jun Yang
* Corresponding author: Jun Yang

Received: May 31, 2017

Revised: August 31, 2017

Published: June 2018

The third author is supported by NSFC(No. 11371254 and No. 11671144)

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Abstract

We consider the Fife-Greenlee problem

$ε^2\triangle u + \bigl(u-\mathbf{a}(y)\bigr)(1-u^2) =0 ~~~ \mbox{in}\ Ω,~~~~~~~\frac{\partial u}{\partialν} = 0 ~~~ \mbox{on}\ \partialΩ,$

where $Ω$ is a bounded domain in ${\mathbb R}^2$ with smooth boundary, $\epsilon>0$ is a small parameter, $ν$ denotes the unit outward normal of $\partialΩ$. Let $Γ = \{y∈ Ω: \mathbf{a}(y) = 0 \}$ be a simple smooth curve intersecting orthogonally with $\partialΩ$ at exactly two points and dividing $Ω$ into two disjoint nonempty components. We assume that $-1\,<\,\mathbf{a}(y)\,<1$ on $Ω$ and $\triangledown\mathbf{a}≠ 0$ on $Γ$, and also some admissibility conditions between the curves $Γ$, $\partialΩ$ and the inhomogeneity ${\mathbf a}$ hold at the connecting points. We can prove that there exists a solution $u_{\epsilon}$ such that: as $\epsilon → 0$, $u_{\epsilon}$ approaches $+1$ in one part, while tends to $-1$ in the other part, except a small neighborhood of $Γ$.

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Mathematics Subject Classification: Primary: 35B34; Secondary: 35J25.

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