题目：Quantitative estimates and asymptotics for fractional Allen-Cahn equations

报告人：王克磊教授, 武汉大学

报告地点：东区第五教学楼5202

报告时间：6月30日, 14:00

摘要：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 address 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 with exceptional regularity—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.