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The strongly nonlocal Allen-Cahn problem

Erisa Hasani, Stefania Patrizi

TL;DR

This work establishes, for the strongly nonlocal regime s in (0,1/2), the sharp interface limit of the fractional Allen–Cahn equation with an L^2-gradient-flow structure. By constructing global and local barriers built from a layer profile φ, a carefully designed corrector ψ, and an auxiliary term ā_ε that encodes fractional mean curvature, the authors prove convergence of u^ε to the minima of the double-well potential and show the interface evolves by fractional mean curvature with velocity proportional to H_{2s}. The analysis extends the viscosity solutions/level-set framework to the s<1/2 regime, addressing substantial nonlocal-elliptic singularities via a new corrector and meticulous estimates, thereby completing the picture alongside prior results for s≥1/2. The results provide a rigorous foundation for nonlocal phase-field models and illuminate the interplay between strong nonlocal diffusion and geometric front dynamics, with potential implications for dislocation theory and nonlocal minimal surfaces.

Abstract

We study the sharp interface limit of the fractional Allen-Cahn equation $$ \varepsilon \partial_t u^{\varepsilon} = \mathcal{I}^s_n [u^{\varepsilon}] -\frac{1}{\varepsilon ^{2s}} W'(u^\varepsilon) \quad \hbox{in}~(0,\infty)\times\mathbb{R}^n, ~n \geq 2, $$ where $\varepsilon >0$, $\mathcal{I}^s_n=-c_{n,s}(-Δ)^s$ is the fractional Laplacian of order $2s\in(0,1)$ in $\mathbb{R}^n$, and $W$ is a smooth double-well potential with minima at 0 and 1. We focus on the singular regime $s\in(0,\frac{1}{2})$, corresponding to strongly nonlocal diffusion. For suitably prepared initial data, we prove that the solution $ u^\varepsilon $ converges, as $\varepsilon\to0$, to the minima of $W$ with the interface evolving by fractional mean curvature flow. This establishes the first rigorous convergence result in this regime, complementing and completing previous work for $s\geq \frac{1}{2}$.

The strongly nonlocal Allen-Cahn problem

TL;DR

This work establishes, for the strongly nonlocal regime s in (0,1/2), the sharp interface limit of the fractional Allen–Cahn equation with an L^2-gradient-flow structure. By constructing global and local barriers built from a layer profile φ, a carefully designed corrector ψ, and an auxiliary term ā_ε that encodes fractional mean curvature, the authors prove convergence of u^ε to the minima of the double-well potential and show the interface evolves by fractional mean curvature with velocity proportional to H_{2s}. The analysis extends the viscosity solutions/level-set framework to the s<1/2 regime, addressing substantial nonlocal-elliptic singularities via a new corrector and meticulous estimates, thereby completing the picture alongside prior results for s≥1/2. The results provide a rigorous foundation for nonlocal phase-field models and illuminate the interplay between strong nonlocal diffusion and geometric front dynamics, with potential implications for dislocation theory and nonlocal minimal surfaces.

Abstract

We study the sharp interface limit of the fractional Allen-Cahn equation where , is the fractional Laplacian of order in , and is a smooth double-well potential with minima at 0 and 1. We focus on the singular regime , corresponding to strongly nonlocal diffusion. For suitably prepared initial data, we prove that the solution converges, as , to the minima of with the interface evolving by fractional mean curvature flow. This establishes the first rigorous convergence result in this regime, complementing and completing previous work for .

Paper Structure

This paper contains 35 sections, 28 theorems, 353 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem and main result
  3. Strategies and prior work.
  4. Future directions
  5. Organization of the paper.
  6. Notations
  7. Motion by fractional mean curvature
  8. The fractional mean curvature
  9. The level set approach
  10. Generalized Flows
  11. Extension of the signed distance function
  12. Heuristics
  13. Derivation of the fractional mean curvature equation
  14. Derivation of equation \ref{['eq:linearized wave']}
  15. Preliminary results on the fractional Laplacian
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $u^\varepsilon = u^\varepsilon(t,x)$ be the unique solution of the reaction-diffusion equation eq:pde with initial datum $u_0^\varepsilon:\mathbb{R}^n \to (0,1)$ defined by where $\phi$ solves eq:standing wave and $d^0$ is given in def:signed_distance_function. Then, as $\varepsilon \to 0$, the solution $u^\varepsilon$ satisfies where $(^+\Omega_t, \Gamma_t, ^-\Omega_t)_{t\geq0}$ denotes the

Figures (2)

  • Figure 1: Initial configuration in dimension $n=2$
  • Figure 2: Convergence result in dimension $n=2$

Theorems & Definitions (45)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4: Extension of the signed distance function
  • Remark 2.5
  • Theorem 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Proposition 4.3: Comparison principle in $\mathbb{R}^n$
  • ...and 35 more