Catenoidal layers for the Allen-Cahn equation in bounded domains

2015/04/21 by Agudelo, O., del Pino, Manuel, Wei, Juncheng
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1504.05301

Abstract

In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation α² Δu + u(1-u²)=0, \hboxin Ω⊂ \R^N where N=3, Ω is a smooth bounded domain and \A>0 is a small parameter. We provide asymptotic behavior which shows that, as α→ 0, the level sets of the solutions collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature that intersects orthogonally ∂ Ω of the domain and that is non-degenerate respect to Ω. We provide explicit examples of surfaces to which our result applies.

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