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arXiv · 2609.30194

Stable solutions of the Allen--Cahn equation in dimension three are one-dimensional

Hardy Chan·Xavier Fernández-Real·Alessio Figalli·Enric Florit-Simon·Joaquim Serra
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Abstract

We prove that every bounded stable solution of the Allen--Cahn equation in $\mathbb R^3$ is one-dimensional. As a consequence, this shows the validity of De Giorgi's conjecture for monotone solutions in dimension four.

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Hardy Chan, Xavier Fernández-Real, Alessio Figalli, Enric Florit-Simon, Joaquim Serra. 2026-09-24. Stable solutions of the Allen--Cahn equation in dimension three are one-dimensional. https://arxiv.org/abs/2609.30194

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