# Transition layer for the heterogeneous Allen–Cahn equation

### Andrea Malchiodi

SISSA, via Beirut 2-4, 34014 Trieste, Italy### Juncheng Wei

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong### Fethi Mahmoudi

SISSA, via Beirut 2-4, 34014 Trieste, Italy

## Abstract

We consider the equation

where $Ω$ is a smooth and bounded domain in $R_{n}$, $ν$ 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 $∇a=0$ on $K$ and $a=0$ on $∂Ω$, we show that there exists a sequence $ɛ_{j}→0$ such that Eq. (1) has a solution $u_{ɛ_{j}}$ which converges uniformly to $±1$ on the compact sets of $Ω_{±}$ as $j→+∞$. This result settles in general dimension a conjecture posed in [P. Fife, M.W. Greenlee, Interior transition layers of elliptic boundary value problem with a small parameter, Russian Math. Surveys 29 (4) (1974) 103–131], proved in [M. del Pino, M. Kowalczyk, J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal. 38 (5) (2007) 1542–1564] only for $n=2$.

## Cite this article

Andrea Malchiodi, Juncheng Wei, Fethi Mahmoudi, Transition layer for the heterogeneous Allen–Cahn equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 3, pp. 609–631

DOI 10.1016/J.ANIHPC.2007.03.008