We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Second-Order Estimates for Transition Layers and a Curvature Estimate for the Parabolic Allen–Cahn.
- Authors
Nguyen, Huy The; Wang, Shengwen
- Abstract
The parabolic Allen–Cahn equation is a semilinear partial differential equation that is closely linked to the mean curvature flow by a singular perturbation. Motivated by the work of Wang–Wei [ 21 ] and Chodosh–Mantoulidis [ 3 ] in the elliptic setting, we initiate the corresponding regularity theory for parabolic Allen–Cahn flows. In particular, we establish an improved convergence property of parabolic Allen–Cahn flows to the mean curvature flow: if the phase-transition level sets converge in |$C^{2}$| , then they converge in |$C^{2,\theta }$| as well. As an application, we obtain a curvature estimate for the parabolic Allen–Cahn equation, which can be seen as a diffused version of Brakke's [ 1 ] and White's [ 24 ] regularity theorems for mean curvature flow.
- Subjects
WANG Wei; CURVATURE; SINGULAR perturbations; PARTIAL differential equations
- Publication
IMRN: International Mathematics Research Notices, 2024, Vol 2024, Issue 8, p6759
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnad269