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$Γ$-convergence for higher order nonlocal phase transitions

Hardy Chan, Serena Dipierro, Mattia Freguglia, Marco Inversi, Enrico Valdinoci

TL;DR

This work investigates Γ-convergence for higher-order nonlocal phase transitions by augmenting the fractional Allen–Cahn energy with the squared L2-norm of its first variation. For all $0<s<3/4$, it is shown that the first-variation term vanishes in the limit, forcing the Γ-limit of the energy pair to be a pure perimeter term: the classical perimeter when $s\ge1/2$ and the $2s$-fractional perimeter when $s<1/2$. The analysis hinges on constructing a 1D optimal-profile-based recovery sequence and a boundary expansion in Fermi coordinates that isolates the curvature contribution, which is shown to vanish in the specified range. Consequently, curvature-dependent terms do not appear in the limit in this regime, contrasting with expectations from local Willmore-type energies and highlighting intrinsic differences between local and nonlocal phase-transition limits. The results refine the understanding of nonlocal Γ-limits and guide expectations for nonlocal geometric energies in low- versus high-regularity regimes.

Abstract

For every $0 < s <3/4$, we study the asymptotic behavior of the $\varepsilon$-rescaled sum of the $s$-fractional Allen-Cahn energy and the squared $L^2$-norm of its first variation. We prove that the contribution of the first variation vanishes as $\varepsilon \to 0$. This implies the Gamma-convergence of the initial sum to either the classical perimeter or to the $2s$-fractional perimeter, depending on whether $s \ge 1/2$ or not. This contradicts the expectation of finding curvature-dependent terms in the limit, as suggested by the regime $3/4 \le s < 1$, and as known to hold in low dimensions in the local case.

$Γ$-convergence for higher order nonlocal phase transitions

TL;DR

This work investigates Γ-convergence for higher-order nonlocal phase transitions by augmenting the fractional Allen–Cahn energy with the squared L2-norm of its first variation. For all , it is shown that the first-variation term vanishes in the limit, forcing the Γ-limit of the energy pair to be a pure perimeter term: the classical perimeter when and the -fractional perimeter when . The analysis hinges on constructing a 1D optimal-profile-based recovery sequence and a boundary expansion in Fermi coordinates that isolates the curvature contribution, which is shown to vanish in the specified range. Consequently, curvature-dependent terms do not appear in the limit in this regime, contrasting with expectations from local Willmore-type energies and highlighting intrinsic differences between local and nonlocal phase-transition limits. The results refine the understanding of nonlocal Γ-limits and guide expectations for nonlocal geometric energies in low- versus high-regularity regimes.

Abstract

For every , we study the asymptotic behavior of the -rescaled sum of the -fractional Allen-Cahn energy and the squared -norm of its first variation. We prove that the contribution of the first variation vanishes as . This implies the Gamma-convergence of the initial sum to either the classical perimeter or to the -fractional perimeter, depending on whether or not. This contradicts the expectation of finding curvature-dependent terms in the limit, as suggested by the regime , and as known to hold in low dimensions in the local case.

Paper Structure

This paper contains 14 sections, 17 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. A brief background
  3. Main result
  4. Heuristic of the proof
  5. Further remarks
  6. Plan of the paper
  7. Tools
  8. The fractional perimeter
  9. The one-dimensional optimal profile
  10. Distance function
  11. On the expansion of the fractional Laplacian around the boundary
  12. Estimates of the four integrals
  13. Proof of Theorem \ref{['t:main']}
  14. The Gamma-convergence result

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^d$ be a bounded Lipschitz domain, and let $E \subset \mathbb{R}^d$ be a Borel set. In both cases, the Gamma-limit is computed with respect to the local $L^1$-topology in $\mathbb{R}^d$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • ...and 21 more