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Non-local planelike minimizers and $Γ$-convergence of periodic energies to a local anisotropic perimeter

Serena Dipierro, Matteo Novaga, Enrico Valdinoci, Riccardo Villa

TL;DR

The paper addresses homogenization of a non-local, periodic interface energy by proving the existence of planelike minimizers via a cell-problem framework and calibrations, and establishing that rescaled energies $\mathscr{F}_\varepsilon$ Γ-converge to a local anisotropic perimeter $\mathscr{F}_\phi$. The stable norm $\phi$ is well-defined, continuous, and captures macroscopic anisotropy as the large-scale limit of microscopic non-local interactions. Key contributions include minimality and density estimates for level sets, controlled oscillations implying slab-like planelike interfaces, and a full Γ-convergence theory comprising both liminf and limsup inequalities. The results connect microscopic non-local dynamics to an effective anisotropic surface energy, enabling rigorous homogenization and paving the way for understanding large-scale interface behavior in periodic media.

Abstract

We investigate a homogenization problem related to a non-local interface energy with a periodic forcing term. We show the existence of planelike minimizers for such energy. Moreover, we prove that, under suitable assumptions on the non-local kernel and the external field, the sequence of rescaled energies $Γ$-converges to a suitable local anisotropic perimeter, where the anisotropy is defined as the limit of the normalized energy of a planelike minimizer in larger and larger cubes (i.e., what is called in jargon "stable norm"). To obtain this, we also establish several auxiliary results, including: the minimality of the level sets of the minimizers, explicit bounds on the oscillations of the minimizers, density estimates for almost minimizers, and non-local perimeter estimates in the large.

Non-local planelike minimizers and $Γ$-convergence of periodic energies to a local anisotropic perimeter

TL;DR

The paper addresses homogenization of a non-local, periodic interface energy by proving the existence of planelike minimizers via a cell-problem framework and calibrations, and establishing that rescaled energies Γ-converge to a local anisotropic perimeter . The stable norm is well-defined, continuous, and captures macroscopic anisotropy as the large-scale limit of microscopic non-local interactions. Key contributions include minimality and density estimates for level sets, controlled oscillations implying slab-like planelike interfaces, and a full Γ-convergence theory comprising both liminf and limsup inequalities. The results connect microscopic non-local dynamics to an effective anisotropic surface energy, enabling rigorous homogenization and paving the way for understanding large-scale interface behavior in periodic media.

Abstract

We investigate a homogenization problem related to a non-local interface energy with a periodic forcing term. We show the existence of planelike minimizers for such energy. Moreover, we prove that, under suitable assumptions on the non-local kernel and the external field, the sequence of rescaled energies -converges to a suitable local anisotropic perimeter, where the anisotropy is defined as the limit of the normalized energy of a planelike minimizer in larger and larger cubes (i.e., what is called in jargon "stable norm"). To obtain this, we also establish several auxiliary results, including: the minimality of the level sets of the minimizers, explicit bounds on the oscillations of the minimizers, density estimates for almost minimizers, and non-local perimeter estimates in the large.
Paper Structure (15 sections, 37 theorems, 408 equations, 3 figures)

This paper contains 15 sections, 37 theorems, 408 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Existence of minimizers for $\text{\Fontauri{E}}_p$ - Proof of Theorem \ref{['th::existence_minimizer']}
  3. Proof of Proposition \ref{['prop::existence_calibration']} and Euler-Lagrange equation for the cell problem \ref{['eq::cell_problem']}
  4. Minimality of level sets - Proof of Theorem \ref{['th::level_sets_minimality']}
  5. Density estimates for minimizers of the functional $\mathscr{F}$
  6. Minimizers of $\text{\Fontauri{E}}_p$ have controlled oscillations - Proof of Theorem \ref{['th::controlled_osc']}
  7. Preliminary results for the $\Gamma$-convergence
  8. Well-definedness of the stable norm $\phi$
  9. Proof of the $\Gamma$-$\liminf$ inequality
  10. Continuity of the stable norm $\phi$
  11. Proof of the $\Gamma$-$\limsup$ inequality
  12. Norms subject to short-range interactions
  13. Uniform density estimates for $\left(\Lambda,\frac{2s_1}{n}\right)$-minimizers of $\operatorname{P}_K$
  14. Behavior of $\operatorname{P}_K$ in large domains
  15. Well-posedness of the Plateau problem for the energy $\mathscr{F}$

Key Result

Theorem 1.4

Let $K:\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}$ satisfy eq::K_invariance, eq::K_integrable, eq::K_behavior, and eq::K_lower_bound_Q. Then, there exists $\gamma>0$, depending only on $n$, $\kappa_1$, $\delta$, and $s_1$, such that, if $\|g\|_{L^{\frac{n}{2s_1}}(Q)}\leqslant\gamma$, there exists

Figures (3)

  • Figure 1: The construction of the set $F'$ as in \ref{['eq::competitor_phi_limit']}.
  • Figure 2: The competitor $F_j$ as in \ref{['eq::competitor_liminf']} that produces the correct $\Gamma-\liminf$ inequality.
  • Figure 3: The planelike approximation of the polygonal set $E$ as in \ref{['eq::poly_approx']}.

Theorems & Definitions (81)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Existence of minimizers for $\text{\Fontauri{E}}_p$
  • Proposition 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Theorem 1.9: Minimality of level sets
  • Theorem 1.10: Minimizers of $\text{\Fontauri{E}}_p$ have controlled oscillations
  • ...and 71 more