Quantitative estimates and asymptotics for fractional Allen-Cahn equation

主讲人：王克磊 武汉大学教授

时间：2026年1月6日16：00

地点：徐汇校区三号楼301室

举办单位：数理学院

主讲人介绍：王克磊，武汉大学数学与统计学院教授， ICCM 数学银奖、ICCM 若琳奖、中国数学会钟家庆奖。
研究方向为非线性偏微分方程、变分法和几何测度论，在 Allen-Cahn 方程、超临界集中现象、DeGiorgi 猜想以及非线性椭圆方程的稳定解和有限 Morse 指标解的分类等问题上做出一系列突出贡献。
在 Bubbling/blow up (Crelle、Poincare、JFA) ；反应扩散方程与行波解 (ARMA、MA.、JMPA)；Allen-Cahn 方程 (CPAM、JEMS、Crelle、Poincare、CPDE、ARMA、AiM、JMPA、TAMS)；有限 Morse 指标 (AiM、MA、CVPDE)；竞争系统 (CPDE、Poincare、AiM、MA、TAMS、JFA) 等高水平期刊发表论文 50 篇。

内容介绍：It is known that the singular limits of Allen-Cahn equations with fractional Laplacian $(-\Delta)^s$ ($0 ＜ s ＜ 1/2$) are nonlocal minimal hypersurfaces (introduced by Caffarelli-Roquejoffre-Savin in [Comm. Pure Appl. Math. 2010]). For energy minimizers, this correspondence was established by Savin and Valdinochi via the $\Gamma$-convergence method.
In this talk, we discuss the case of general critical points. By using the quantitative stratification theory developed by Cheeger-Naber and Naber-Valtorta, we derive precise estimates for transition layers in fractional Allen-Cahn equations. These estimates imply that general critical points converge to nonlocal minimal hypersurfaces, which is substantially better than the convergence behavior observed in classical Allen-Cahn equations.
The talk is based on two joint works with Vincent Millot and Yannick Sire, Juncheng Wei and Ke Wu.