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$Γ$-convergence for nonlocal phase transitions involving the $H^{1/2}$ norm

Tim Heilmann

TL;DR

This work investigates nonlocal phase-transition energies $F_\varepsilon$ combining a double-well potential and a Gagliardo $H^{1/2}$–norm interaction, under the scaling $\varepsilon \log \lambda_\varepsilon \to k$. The authors prove compactness in $BV(\Omega;\{\alpha,\beta\})$ and establish $\Gamma$-convergence to an isotropic perimeter functional $F(u)=2(\beta-\alpha)^2 \omega_{n-1} k \mathcal{H}^{N-1}(S_u)$, with finite energy only on BV functions attaining the wells. The proof blends sharp nonlocal lower bounds (via cylinder and hyperplane geometries) with a localization and reconstruction strategy to build recovery sequences on polyhedral interfaces. The results extend nonlocal phase-transition theory to exponential-well growth regimes and yield an isotropic surface-energy limit independent of kernel anisotropy.

Abstract

We study functionals \begin{equation*} F_\varepsilon (u) := λ_\varepsilon \int_ΩW(u) \, dx + \varepsilon \|u\|_{H^{1/2}}^2 \end{equation*} for a double well potential $W$ and the Gagliardo seminorm $\|\cdot\|_{H^{1/2}}$ when $\varepsilon \ln(λ_\varepsilon) \rightarrow k$ as $\varepsilon \rightarrow 0^+$ and show compactness in the space of $BV$ functions on $Ω$ and the $Γ$-convergence to the classical surface tension functional.

$Γ$-convergence for nonlocal phase transitions involving the $H^{1/2}$ norm

TL;DR

This work investigates nonlocal phase-transition energies combining a double-well potential and a Gagliardo –norm interaction, under the scaling . The authors prove compactness in and establish -convergence to an isotropic perimeter functional , with finite energy only on BV functions attaining the wells. The proof blends sharp nonlocal lower bounds (via cylinder and hyperplane geometries) with a localization and reconstruction strategy to build recovery sequences on polyhedral interfaces. The results extend nonlocal phase-transition theory to exponential-well growth regimes and yield an isotropic surface-energy limit independent of kernel anisotropy.

Abstract

We study functionals \begin{equation*} F_\varepsilon (u) := λ_\varepsilon \int_ΩW(u) \, dx + \varepsilon \|u\|_{H^{1/2}}^2 \end{equation*} for a double well potential and the Gagliardo seminorm when as and show compactness in the space of functions on and the -convergence to the classical surface tension functional.

Paper Structure

This paper contains 9 sections, 13 theorems, 120 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. The energy functional, the main theorem and proof ideas
  4. Proof of the main result
  5. Estimates for the nonlocal energy
  6. Lower bounds for the nonlocal energy
  7. Energy bounds on cubes
  8. Compactness
  9. $\Gamma$-convergence

Key Result

Theorem 1.1

If $F_\varepsilon, F$ and $W$ are as defined above, the following statements hold: Compactness: Given sequences $(u_{\varepsilon_h})_h$ in $L^1(\Omega)$ and $(\varepsilon_h)_h$ in $\mathop{\mathrm{\mathbb{R}}}\nolimits$ such that $\varepsilon_h \rightarrow 0^+$ and $F_{\varepsilon_h}(u_{\varepsilon_

Figures (5)

  • Figure 1: Shifts of the cylinder $A$ allow to reduce to the situation of Lemma \ref{['planeplane']}
  • Figure 2: The decomposition of $A$ and $B$ from the proof of Lemma \ref{['cylindercomplement']} and the set $\mathcal{R}$ from Remark \ref{['cylindercone']}
  • Figure 3: The sets $D$ and $\hat{B}$ from Lemma \ref{['cylindercomplement']}
  • Figure 4: Example of the situation of Lemma \ref{['correction_bound']} for $i = 0$ and $j = 2$
  • Figure 5: Example of the situation of Corollary \ref{['correction_cylinderbound2']}; the density of $A \cup B$ is assumed to be big

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1: Computation of G
  • Lemma 2.2: Value of $H$ on a line for two fattened hyperplanes
  • proof
  • Lemma 2.3: Value of $G$ for a cylinder and a fattened hyperplane
  • proof
  • Lemma 2.4: Estimate of $G$ for a cylinder and its complement inside a fattened hyperplane
  • proof
  • Remark 2.5: Estimate of $G$ for a cylinder and the complement of a cone inside a fattened hyperplane
  • ...and 18 more