Boundary interface for the Allen–Cahn equation

Abstract.

We consider the Allen–Cahn equation

$$ \varepsilon^{2}\Delta u + u - u^3 = 0 \quad {\rm in}\,\Omega, \quad \frac{\partial u}{\partial v} = 0 \quad {\rm on}\,\partial\Omega, $$

where Ω is a smooth and bounded domain in ${\mathbb{R}}^n$ such that the mean curvature is positive at each boundary point. We show that there exists a sequence ε _j → 0 such that the Allen–Cahn equation has a solution $u_{\varepsilon_j}$ with an interface which approaches the boundary as j → + ∞.