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arXiv · 2610.01335

The Allen-Cahn equation and general minimal cones

Oscar Ivan Agudelo Rico·Matteo Rizzi
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Abstract

We construct solutions to the Allen-Cahn equation $Δu+u-u^3=0$ in $\mathbb{R}^{N+1}$, with $N\ge 3$, whose zero level set is asymptotic (at infinity) to a given minimal cone. Our construction is quite general, since we only use a suitable nondegeneracy assumption; neither stability nor symmetry of the underlying cone is required. Moreover, we calculate the morse index of such solutions. In particular, our main result generalizes a result by Pacard and Wei (2013) dealing with stable solutions to the Allen-Cahn equation in high dimensions. More precisely, we give a positive answer to a remark raised in the paper by Pacard and Wei about the possibility to relax the assumption for the underlying cone is area-minimizing or at least stable. As a corollary, we also obtain solutions with infinite morse index and connected zero level set.

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Oscar Ivan Agudelo Rico, Matteo Rizzi. 2026-10-01. The Allen-Cahn equation and general minimal cones. https://arxiv.org/abs/2610.01335

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