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

TL;DR

This paper constructs multi-layered solutions to an inhomogeneous Allen-Cahn equation in a bounded domain, showing layers cluster near the boundary with specific spacing, under curvature and resonance conditions.

Contribution

It extends previous work by establishing clustered boundary-layer solutions for the inhomogeneous Allen-Cahn equation in two dimensions with detailed geometric conditions.

Findings

Existence of N-layer solutions near boundary with mutual distance O(ε|ln ε|)

Solutions exist when boundary curvature is positive and away from resonance values

Generalizes prior results to inhomogeneous case with boundary curvature considerations.

Abstract

We consider the inhomogeneous Allen-Cahn equation $$ \epsilon^2\Delta u\,+\,V(y)(1-u^2)\,u\,=\,0\quad \mbox{in}\ \Omega, \qquad \frac {\partial u}{\partial \nu}\,=\,0\quad \mbox{on}\ \partial \Omega, $$ 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 $N\geq 2$, 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)