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arXiv · 1504.05301

Catenoidal layers for the Allen-Cahn equation in bounded domains

O. Agudelo·Manuel del Pino·Juncheng Wei
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Abstract

In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation $α^2 Δu + u(1-u^2)=0, \quad \hbox{in }Ω\subset \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 $α\to 0$, the level sets of the solutions collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature that intersects orthogonally $\partial Ω$ 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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O. Agudelo, Manuel del Pino, Juncheng Wei. 2015-04-21. Catenoidal layers for the Allen-Cahn equation in bounded domains. https://arxiv.org/abs/1504.05301

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