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

Abstract

We consider the inhomogeneous Allen-Cahn equation $$ ε^2Δu\,+\,V(y)(1-u^2)\,u\,=\,0\quad \mbox{in}\ Ω, \qquad \frac {\partial u}{\partial ν}\,=\,0\quad \mbox{on}\ \partial Ω, $$ where $\Omega$ is a bounded domain in ${\mathbb{R}}^2$ with smooth boundary $\partial \Omega$ and $V(x)$ is a positive smooth function, $\epsilon >0$ is a small parameter, $\nu$ denotes the unit outward normal of $\partial \Omega$. For any fixed integer $N\ge 2$, we will show the existence of a clustered solution $u_{\epsilon }$ with $N$-transition layers near $\partial \Omega$ with mutual distance $O(\epsilon |\ln \epsilon |)$, provided that the generalized mean curvature $\mathcal{H}$ of $\partial \Omega$ is positive and $\epsilon$ stays away from a discrete set of values at which resonance occurs. Our result is an extension of those (with dimension two) by A. Malchiodi, W.-M. Ni, J. Wei in Pacific J. Math. (Vol. 229, 2007, no. 2, 447-468) and A. Malchiodi, J. Wei in J. Fixed Point Theory Appl. (Vol. 1, 2007, no. 2, 305-336)