Second order estimates on transition layers

TL;DR

This paper proves a uniform second-order regularity estimate for level sets of stable solutions to the Allen-Cahn equation in dimensions up to 10, combining reduction to Toda systems and decay estimates.

Contribution

It introduces a novel approach linking Allen-Cahn solutions to Toda systems to establish optimal regularity estimates.

Findings

Established $C^{2, heta}$ estimates for level sets in dimensions $n\, ext{≤}\,10$

Reduced the problem to Toda system solutions using infinite dimensional reduction

Derived decay estimates that bound distances between solution sheets

Abstract

In this paper we establish a uniform $C^{2,\theta}$ estimate for level sets of stable solutions to the singularly perturbed Allen-Cahn equation in dimensions $ n\leq 10$ (which is optimal). The proof combines two ingredients: one is the infinite dimensional reduction method which enables us to reduce the $C^{2,\theta}$ 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.