Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative stratification for the fractional Allen-Cahn equation and stationary nonlocal minimal surface

Kelei Wang, Juncheng Wei, Ke Wu

TL;DR

The paper addresses sharp quantitative control of phase interfaces for the fractional Allen-Cahn equation with $s\in(0,\tfrac{1}{2})$ in $n\ge2$ dimensions. By adapting the quantitative stratification framework of Naber–Valtorta to the nonlocal setting via the Dirichlet-to-Neumann extension, it derives a corona-type decomposition and sharp bounds for the transition set and energy. The main contributions are optimal tubular-volume bounds on the transition region, precise convergence rates for the potential energy depending on $s$, and optimal $2s$-perimeter bounds for stationary nonlocal minimal surfaces, extending previous results without requiring strong energy assumptions. These results advance the understanding of nonlocal phase interfaces and their geometric regularity, with implications for the Gamma-convergence to nonlocal minimal surfaces and their rectifiability.

Abstract

We study properties of solutions to the fractional Allen-Cahn equation when $s\in (0, 1/2)$ and dimension $n\geq 2$. By applying the quantitative stratification principle developed by Naber and Valtorta, we obtain an optimal quantitative estimate on the transition set. As an application of this estimate, we improve the potential energy estimates of Cabre, Cinti, and Serra (2021), providing sharp versions for the fractional Allen-Cahn equation. Similarly, we obtain optimal perimeter estimates for stationary nonlocal minimal surfaces, extending previous results of Cinti, Serra, and Valdinoci (2019) from the stable case.

Quantitative stratification for the fractional Allen-Cahn equation and stationary nonlocal minimal surface

TL;DR

The paper addresses sharp quantitative control of phase interfaces for the fractional Allen-Cahn equation with in dimensions. By adapting the quantitative stratification framework of Naber–Valtorta to the nonlocal setting via the Dirichlet-to-Neumann extension, it derives a corona-type decomposition and sharp bounds for the transition set and energy. The main contributions are optimal tubular-volume bounds on the transition region, precise convergence rates for the potential energy depending on , and optimal -perimeter bounds for stationary nonlocal minimal surfaces, extending previous results without requiring strong energy assumptions. These results advance the understanding of nonlocal phase interfaces and their geometric regularity, with implications for the Gamma-convergence to nonlocal minimal surfaces and their rectifiability.

Abstract

We study properties of solutions to the fractional Allen-Cahn equation when and dimension . By applying the quantitative stratification principle developed by Naber and Valtorta, we obtain an optimal quantitative estimate on the transition set. As an application of this estimate, we improve the potential energy estimates of Cabre, Cinti, and Serra (2021), providing sharp versions for the fractional Allen-Cahn equation. Similarly, we obtain optimal perimeter estimates for stationary nonlocal minimal surfaces, extending previous results of Cinti, Serra, and Valdinoci (2019) from the stable case.

Paper Structure

This paper contains 16 sections, 30 theorems, 251 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. The Dirichlet-to-Neumann Operator
  5. Preliminaries
  6. Symmetry of functions
  7. Reifenberg type theorems
  8. The $L^{2}$ best approximation estimate
  9. Centered density drop gives packing
  10. The key dichotomy
  11. A corona-type decomposition
  12. Good tree construction
  13. Bad tree construction
  14. Alternating the trees
  15. The proof of of Theorem \ref{['themain1']}
  16. ...and 1 more sections

Key Result

Theorem 1.3

Let $\{g_{\epsilon}\}_{\epsilon\in (0, 1)}\subset C_{loc}^{0, 1}(\mathbb{R}^{n})$ be such that and $g_{\epsilon}\to g$ strongly in $L_{loc}^{1}(\mathbb{R}^{n}\backslash B_{40}(0))$ for a function $g$ satisfying $|g|=1$ almost everywhere in $\mathbb{R}^{n}\backslash B_{40}(0)$. For each $\epsilon\in (0, 1)$, let $u_{\epsilon}\in H^{s}_{g_{\epsilon}}(B_{40}(0))\cap L^{\infty}(\mathbb{R}^{n})$ b and

Theorems & Definitions (64)

  • Definition 1.1: Weak solutions
  • Definition 1.2: Stationary nonlocal minimal surfaces
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 54 more