Allen--Cahn方程在$\mathbb{R}^3$中的稳定De Giorgi猜想
Stable De Giorgi conjecture of the Allen--Cahn equation in $\mathbb{R}^3$
- Beijing Technology and Business University（北京工商大学）
- University of Science and Technology of China（中国科学技术大学）
- Wuhan University（武汉大学）
- Chinese University of Hong Kong（香港中文大学）
- Yunnan Normal University（云南师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
证明$\mathbb{R}^3$中Allen--Cahn方程的有界稳定整体解必为一维，进而推出$\mathbb{R}^4$中完整De Giorgi猜想成立；方法是将稳定性条件归结为零集上的弱稳定性条件并利用Gauss-Bonnet公式构造矛盾。
AI中文摘要：
我们证明了$\mathbb{R}^3$中Allen--Cahn方程的每个有界稳定整体解$v$都是一维的。作为推论，$\mathbb{R}^4$中的完整De Giorgi猜想成立。我们还获得了稳定解的局部曲率估计。证明策略受到Chan、Fernández-Real、Figalli和Serra近期突破性工作[J. Amer. Math. Soc. 2026]的启发，通过将Allen--Cahn方程的稳定性条件归结为曲面（零集）上的弱稳定性条件，然后利用Gauss-Bonnet公式。为此，我们首先利用稳定性条件获得一个加权积分的次线性界，该积分控制解远离平面的零点。如果这样的零点存在，我们隔离出其中的一个有界集合，并将该集合附近的$1-v^2$与更远处一维跃迁的导数连接起来。通过控制这些跃迁之间的相互作用，我们推导出零集上的弱稳定性条件，然后用该条件将零集正则部分上曲率平方的加权积分用截断梯度积分和受控误差来界定。我们利用该不等式来界定内蕴面积并构造对数截断。所得紧支撑测试函数在该集合附近产生负贡献，该负贡献超过所有连接和截断误差，从而与稳定性矛盾。
英文摘要：
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 $\mathbb{R}^4$ 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.
