Catenoidal layers for the Allen-Cahn equation in bounded domains

Abstract

This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α ²Δu + u(1 − u ²) = 0 in a smooth bounded domain Ω ⊂ R³, with Neumann boundary condition and α > 0 a small parameter. These solutions have the property that as α → 0, their level sets collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature intersecting ∂Ω orthogonally and that is non-degenerate respect to ∂Ω. The authors provide explicit examples of surfaces to which the result applies.

Keywords

Allen-Cahn equation / Critical minimal surfaces / Critical catenoid / Infinite dimensional gluing method / Neumann boundary condition

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