2018/10/22 by Kelei Wang, Juncheng Wei, Wang, Kelei +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1810.09599
openalex publication_date 2018/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we establish a uniform C2,θ estimate for level sets of stable solutions to the singularly perturbed Allen-Cahn equation in dimensions n≤ 10 (which is optimal). The proof combines two ingredients: one is the infinite dimensional reduction method which enables us to reduce the C2,θ estimate for these level sets to a corresponding one on solutions of Toda system; the other one uses a small regularity theorem on stable solutions of Toda system to establish various decay estimates on these solutions, which gives a lower bound on distances between different sheets of solutions to Toda system or level sets of solutions to Allen-Cahn equation.