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A rigorous approach to pattern formation for isotropic isoperimetric problems with competing nonlocal interactions

Sara Daneri, Eris Runa

TL;DR

This work develops a rigorous calculus-of-variations framework for isotropic, nonlocal isoperimetric energies in dimensions $d\ge2$, balancing a local perimeter term against a power-law nonlocal kernel. By introducing an integral-geometric decomposition and deriving sharp 1D estimates, the authors establish a rigidity mechanism that forces minimizers to adopt stripe-like, flat-boundary configurations in the critical regime where the competing terms are co-dominant. A nonlocal curvature density is identified, which tightly controls boundary regularity and, under suitable decay, enforces flat interfaces; a detailed regularity theory shows that finite-energy configurations are Lipschitz and, in stronger decay regimes, planar. Through a 2D slicing argument, the 2D rigidity results extend to all dimensions, yielding exact striped minimizers for small parameter perturbations and providing a Gamma-convergence description of the energy to stripe-dominated limits. The findings illuminate pattern formation in isotropic nonlocal systems and connect to physical contexts such as synthetic antiferromagnets, where stripe-like modulated phases are expected in the critical regime.

Abstract

We introduce a rigorous approach to the study of the symmetry breaking and pattern formation phenomenon for isotropic functionals with local/nonlocal interactions in competition. We consider a general class of nonlocal variational problems in dimension $d\geq 2$, in which an isotropic surface term favouring pure phases competes with an isotropic nonlocal term with power law kernel favouring alternation between different phases. Close to the critical regime in which the two terms are of the same order, we give a rigorous proof of the conjectured structure of global minimizers, in the shape of domains with flat boundary (e.g., stripes or lamellae). The natural framework in which our approach is set and developed is the one of calculus of variations and geometric measure theory. Among others, we detect a nonlocal curvature-type quantity which is controlled by the energy functional and whose finiteness implies flatness for sufficiently regular boundaries. The power of decay of the considered kernels at infinity is $p\geq d+3$ and it is related to pattern formation in synthetic antiferromagnets.

A rigorous approach to pattern formation for isotropic isoperimetric problems with competing nonlocal interactions

TL;DR

This work develops a rigorous calculus-of-variations framework for isotropic, nonlocal isoperimetric energies in dimensions , balancing a local perimeter term against a power-law nonlocal kernel. By introducing an integral-geometric decomposition and deriving sharp 1D estimates, the authors establish a rigidity mechanism that forces minimizers to adopt stripe-like, flat-boundary configurations in the critical regime where the competing terms are co-dominant. A nonlocal curvature density is identified, which tightly controls boundary regularity and, under suitable decay, enforces flat interfaces; a detailed regularity theory shows that finite-energy configurations are Lipschitz and, in stronger decay regimes, planar. Through a 2D slicing argument, the 2D rigidity results extend to all dimensions, yielding exact striped minimizers for small parameter perturbations and providing a Gamma-convergence description of the energy to stripe-dominated limits. The findings illuminate pattern formation in isotropic nonlocal systems and connect to physical contexts such as synthetic antiferromagnets, where stripe-like modulated phases are expected in the critical regime.

Abstract

We introduce a rigorous approach to the study of the symmetry breaking and pattern formation phenomenon for isotropic functionals with local/nonlocal interactions in competition. We consider a general class of nonlocal variational problems in dimension , in which an isotropic surface term favouring pure phases competes with an isotropic nonlocal term with power law kernel favouring alternation between different phases. Close to the critical regime in which the two terms are of the same order, we give a rigorous proof of the conjectured structure of global minimizers, in the shape of domains with flat boundary (e.g., stripes or lamellae). The natural framework in which our approach is set and developed is the one of calculus of variations and geometric measure theory. Among others, we detect a nonlocal curvature-type quantity which is controlled by the energy functional and whose finiteness implies flatness for sufficiently regular boundaries. The power of decay of the considered kernels at infinity is and it is related to pattern formation in synthetic antiferromagnets.
Paper Structure (10 sections, 22 theorems, 233 equations, 4 figures)

This paper contains 10 sections, 22 theorems, 233 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Notation and preliminaries
  4. An integral geometric formulation
  5. One dimensional estimates
  6. A rigidity result
  7. (i) A nonlocal curvature bound
  8. (iii) Regularity and nonlocal curvature bounds imply flatness
  9. (ii) Regularity of sets of finite energy
  10. (iv) Rigidity: from $d=2$ to arbitrary dimension

Key Result

Theorem 1.1

Let $d\geq 1$, $p\geq d+3$, $L>0$. Then, there exists $\hat{\tau}>0$ such that for every $0<\tau < \hat{\tau}$ the $[0,L)^d$-periodic minimizers $E_\tau$ of $\mathcal{F}_{\tau,p,d}(\cdot, [0,L)^d)$ are, up to a rigid motion, of the form for some $h^*_{L}>0$ such that $2kh^*_{L}=L$ for some $k\in\mathbb N$.

Figures (4)

  • Figure 1: The light grey regions correspond to the sets $C^+(x,\|y-x\|)\cap C^-(y,\|y-x\|)$, $C^-(x,\|y-x\|)\cap C^+(y,\|y-x\|)$.
  • Figure 2: By the blow-up properties \ref{['eq:stimeblowup']}, a difference between $\nu_E(x)$ and $\nu_E(y)$ controls from below $\Omega(x,\|y-x\|)$ and $\Omega(y,\|y-x\|)$.
  • Figure 3: By the blow-up properties \ref{['eq:stimeblowup']}, if the set $\{\theta\in\mathbb{S}^{d-1}:\,\mathbf{r}_\theta(z)<r\}$ has small measure, then the set $E\cap B_r(z)$ is close to $H_{\nu_E(z)}(z)\cap B_r(z)$
  • Figure 4: A picture for Case 2 of Lemma \ref{['lemma:excf']}, at a point $x$ such that $|\langle \nu_E(x) , \bar{\theta} \rangle|\ll1$.

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Proposition 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 40 more