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On the shape of minimizers for the periodic nonlocal perimeter in $\mathbb{R}^2$

Renzo Bruera

TL;DR

This work investigates the stability and shape of minimizers for the periodic nonlocal perimeter in 2D. By combining bifurcation theory from straight cylinders with a detailed spectral analysis of the linearized nonlocal mean curvature operator, the authors construct sequences of periodic constant-NMC near-cylinders (near-bands in 2D) and derive a Rayleigh-quotient criterion that governs stability. They prove that, for sufficiently smooth and flat profiles, any non-straight near-cylinder is unstable under even, cylindrically symmetric, volume-preserving periodic variations, and they establish that straight cylinders with radius at least $R_1$ are stable, while smaller radii are unstable. Consequently, at large volumes the periodic nonlocal isoperimetric minimizers are conjectured to be straight bands, paralleling the local case, with precise asymptotics as $\alpha\uparrow1$ reinforcing the limiting behavior. The analysis hinges on explicit eigenvalue computations $\mu_k(R)$, bifurcation radii $R_m=R_1/m$, and careful asymptotics of the second variation, yielding a rigorous picture of when nontrivial periodic minimizers can occur in the nonlocal setting.

Abstract

In this paper, we study planar nonlocal Delaunay sets. That is, open sets in $\mathbb{R}^2$ with constant nonlocal mean curvature that are periodic in $x_1$, and even in $x_1$ and in $x_2$. Using bifurcation analysis and fine explicit computations, we prove that every sufficiently $C^{1,β}$-flat nonlocal Delaunay set in $\mathbb{R}^2$ that is not a straight band is unstable with respect to volume-preserving periodic variations. Our results support the conjecture that, as in the local case, in the range of large areas, minimizers of the periodic nonlocal isoperimetric problem -- also known as the nonlocal liquid drop problem with prescribed area between two parallel hyperplanes -- are all straight bands.

On the shape of minimizers for the periodic nonlocal perimeter in $\mathbb{R}^2$

TL;DR

This work investigates the stability and shape of minimizers for the periodic nonlocal perimeter in 2D. By combining bifurcation theory from straight cylinders with a detailed spectral analysis of the linearized nonlocal mean curvature operator, the authors construct sequences of periodic constant-NMC near-cylinders (near-bands in 2D) and derive a Rayleigh-quotient criterion that governs stability. They prove that, for sufficiently smooth and flat profiles, any non-straight near-cylinder is unstable under even, cylindrically symmetric, volume-preserving periodic variations, and they establish that straight cylinders with radius at least are stable, while smaller radii are unstable. Consequently, at large volumes the periodic nonlocal isoperimetric minimizers are conjectured to be straight bands, paralleling the local case, with precise asymptotics as reinforcing the limiting behavior. The analysis hinges on explicit eigenvalue computations , bifurcation radii , and careful asymptotics of the second variation, yielding a rigorous picture of when nontrivial periodic minimizers can occur in the nonlocal setting.

Abstract

In this paper, we study planar nonlocal Delaunay sets. That is, open sets in with constant nonlocal mean curvature that are periodic in , and even in and in . Using bifurcation analysis and fine explicit computations, we prove that every sufficiently -flat nonlocal Delaunay set in that is not a straight band is unstable with respect to volume-preserving periodic variations. Our results support the conjecture that, as in the local case, in the range of large areas, minimizers of the periodic nonlocal isoperimetric problem -- also known as the nonlocal liquid drop problem with prescribed area between two parallel hyperplanes -- are all straight bands.
Paper Structure (16 sections, 17 theorems, 208 equations, 2 figures)

This paper contains 16 sections, 17 theorems, 208 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. A sequence of bifurcated families of Delaunay near-cylinders in $\mathbb{R}^2$
  4. On the stability of periodic constant-NMC surfaces of revolution
  5. A Rayleigh quotient criterion for stability
  6. The case of straight cylinders in dimension 2
  7. On the eigenvalues of the linearized NMC operator
  8. A general expression for $\ddot{\mu}_1$ in smooth families of operators
  9. The case of the linearized NMC operator
  10. Continuation of the eigenvalues
  11. On the stability of the near-cylinders
  12. An asymptotic formula for $\mathcal{R}_1(a)$
  13. Instability of the near-cylinders
  14. Computations for the Rayleigh quotient on the near-cylinders
  15. Proof of Proposition \ref{['proposition:dotmu_ddotmu_dote1']}
  16. ...and 1 more sections

Key Result

Theorem 1.3

Let $n=2$, $\alpha\in (0,1)$ and $\alpha< \beta < \min\{1, 2\alpha+\frac{1}{2}\}$. There exists $R_1>0$, depending only on $\alpha$, such that, for every $m\geq 1$, denoting $R_m=R_1/m$, the pair $(R_m,u_{R_m})$ is a bifurcation point of the equation As a consequence, for every $m\geq 1$ there exists $\nu_m>0$, depending only on $\alpha$, $\beta$, and $m$, and a smooth curve of nontrivial solutio

Figures (2)

  • Figure 1: Representation of the space $C^{1,\beta}_{even}(\mathbb{T}^1)$ decomposed into $X_0$ (the subspace of constant functions) and $X_0^\perp$ (the subspace of functions orthogonal to $1$ in $L^2_{even}(\mathbb{T}^1)$). The horizontal axis corresponds to straight cylinders (that is, the trivial solutions $u_R$), and the vertical curves at each $u_{R_m}$ correspond to the near-cylinders of Theorem \ref{['theorem_seq_bif']}. As stated in Theorem \ref{['theorem_stability_near_cylinders']}, straight cylinders lying to the right of $u_{R_1}$ are stable, whereas those lying to its left are unstable. The dashed bifurcated curves are all unstable. Corollary \ref{['cor::classification_minimizers']} states that all critical points different from straight cylinders lying in the two shaded areas in the Figure are unstable when $\alpha\geq \alpha_0$. When $\alpha<\alpha_0$, we expect the same result to hold, but we can only rigorously conclude their instability in the lighter shaded area. Notice the pitchfork-like bifurcation occurring at $u_{R_1}$.
  • Figure 2: Representation of the values of $R_1$ and $\ddot{\mathcal{R}}_1$ as a function of $\alpha$ obtained by numerical computation of the function in \ref{['expression_ddot_mu_hom_kernel']}. Since $\ddot{\mathcal{R}}_1(0)$ diverges quadratically as $\alpha\uparrow 1$, we represent its product with $(1-\alpha)^2$. Notice how $\ddot{\mathcal{R}}_1(0)<0$ for all $\alpha\in (0,1)$.

Theorems & Definitions (41)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Lemma 2.1: CNLMCDelCyl
  • Lemma 2.2: CabreFallWeth2018
  • Lemma 2.3: CNLMCDelCyl
  • ...and 31 more