Clustering of boundary interfaces for an inhomogeneous Allen–Cahn equation on a smooth bounded domain

Abstract

We consider the inhomogeneous Allen–Cahn equation $$\begin{aligned} \epsilon ^2\Delta u\,+\,V(y)(1-u^2)\,u\,=\,0\quad \text{ in }\ \Omega , \qquad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\ \partial \Omega , \end{aligned}$$ 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 Malchiodi et al. (Pac. J. Math. 229(2):447–468, 2007) and Malchiodi et al. (J. Fixed Point Theory Appl. 1(2):305–336, 2007).