Stable De Giorgi conjecture of the Allen--Cahn equation in ℝ3
Abstract
We prove that every bounded stable entire solution v of the Allen--Cahn equation in R3 is one-dimensional. As a consequence, the full De Giorgi conjecture in ℝ4 is true. We also obtain local curvature estimates for stable solutions. The proof strategy is inspired by the recent breakthrough work of Chan, Fernández-Real, Figalli and Serra [J. Amer. Math. Soc. 2026], by reducing the stabilty condition for the Allen-Cahn equation to a weak stability condition on a surface (the zero set) and then utilizing Gauss-Bonnet formula. For this purpose, we first use the stability condition to get a sublinear bound for a weighted integral that controls the zeros where the solution is far from planar. If such zeros exist, we isolate a bounded set of them and join 1−v2 near this set to derivatives of one-dimensional transitions farther away. By controlling the interaction between these transitions, we derive the weak stability condition on the zero set, which is then used to bound a weighted integral of the squared curvature on the regular part of the zero set by a cutoff gradient integral and a controlled error. We use this inequality to bound the intrinsic area and construct logarithmic cutoffs. The resulting compactly supported test function has a negative contribution near this set that exceeds all joining and cutoff errors, contradicting stability.
Related
- Stable solutions of the Allen--Cahn equation in dimension three are one-dimensional
- Sheeting property and rigidity for stable solutions to the Allen--Cahn equation
- Multiplicity One for Strictly Stable Anisotropic Allen-Cahn Minimal Hypersurfaces
- A Half-Space Theorem for the Allen--Cahn Equation in All Dimensions
- Finite Morse index solutions of the Allen-Cahn equation with energy bound
BibTeX
@misc{liu2026stable,
title = {{Stable De Giorgi conjecture of the Allen--Cahn equation in $\mathbbR^3$}},
author = {Yong Liu and Tianci Luo and Kelei Wang and Juncheng Wei and Yong Wei and Ke Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21680},
eprint = {2609.21680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21680},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Stable De Giorgi conjecture of the Allen--Cahn equation in ℝ^3
AU - Yong Liu
AU - Tianci Luo
AU - Kelei Wang
AU - Juncheng Wei
AU - Yong Wei
AU - Ke Wu
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21680
UR - https://arxiv.org/abs/2609.21680
AB - We prove that every bounded stable entire solution v of the Allen--Cahn equation in R^3 is one-dimensional. As a consequence, the full De Giorgi conjecture in ℝ^4 is true. We also obtain local curvature estimates for stable solutions. The proof strategy is inspired by the recent breakthrough work of Chan, Fernández-Real, Figalli and Serra [J. Amer. Math. Soc. 2026], by reducing the stabilty condition for the Allen-Cahn equation to a weak stability condition on a surface (the zero set) and then utilizing Gauss-Bonnet formula. For this purpose, we first use the stability condition to get a sublinear bound for a weighted integral that controls the zeros where the solution is far from planar. If such zeros exist, we isolate a bounded set of them and join 1−v^2 near this set to derivatives of one-dimensional transitions farther away. By controlling the interaction between these transitions, we derive the weak stability condition on the zero set, which is then used to bound a weighted integral of the squared curvature on the regular part of the zero set by a cutoff gradient integral and a controlled error. We use this inequality to bound the intrinsic area and construct logarithmic cutoffs. The resulting compactly supported test function has a negative contribution near this set that exceeds all joining and cutoff errors, contradicting stability.
ER -