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On the existence of minimizing sets for a weakly-repulsive non-local energy

Davide Carazzato, Aldo Pratelli, Ihsan Topaloglu

TL;DR

The paper investigates minimizing sets for a weakly repulsive nonlocal energy defined by a radial kernel $g$. By bridging density minimizers and measure minimizers, it proves the existence of optimal sets for small mass $m$ under general regularity assumptions on $g$, and specializes to power-law kernels to obtain precise structural results: for suitable parameters, minimizers exist as sets consisting of $N{+}1$ convex components near the vertices of a unit regular $(N{+}1)$-gon, while in other regimes the minimizers are radial annuli or disks for all masses. It further leverages results of Davies–Lim–McCann to show that when the limiting measure concentrates on the $\Delta_N$ vertices, set minimizers exist for small $m$, and when the limiting measure is sphere-supported, minimizers exist for all $m$ with annular or disk geometry. A planar geometric confinement argument also yields conditions under which the unique minimizer is atomic on the $\Delta_N$ vertices, with higher-dimensional extensions discussed.

Abstract

We consider a non-local interaction energy over bounded densities of fixed mass $m$. We prove that under certain regularity assumptions on the interaction kernel these energies admit minimizers given by characteristic functions of sets when $m$ is sufficiently small (or even for every $m$, in particular cases). We show that these assumptions are satisfied by particular interaction kernels in power-law form, and give a certain characterization of minimizing sets. Finally, following a recent result of Davies, Lim and McCann, we give sufficient conditions on the interaction kernel so that the minimizer of the energy over probability measures is given by Dirac masses concentrated on the vertices of a regular $(N+1)$-gon of side length 1 in $\mathbb{R}^N$.

On the existence of minimizing sets for a weakly-repulsive non-local energy

TL;DR

The paper investigates minimizing sets for a weakly repulsive nonlocal energy defined by a radial kernel . By bridging density minimizers and measure minimizers, it proves the existence of optimal sets for small mass under general regularity assumptions on , and specializes to power-law kernels to obtain precise structural results: for suitable parameters, minimizers exist as sets consisting of convex components near the vertices of a unit regular -gon, while in other regimes the minimizers are radial annuli or disks for all masses. It further leverages results of Davies–Lim–McCann to show that when the limiting measure concentrates on the vertices, set minimizers exist for small , and when the limiting measure is sphere-supported, minimizers exist for all with annular or disk geometry. A planar geometric confinement argument also yields conditions under which the unique minimizer is atomic on the vertices, with higher-dimensional extensions discussed.

Abstract

We consider a non-local interaction energy over bounded densities of fixed mass . We prove that under certain regularity assumptions on the interaction kernel these energies admit minimizers given by characteristic functions of sets when is sufficiently small (or even for every , in particular cases). We show that these assumptions are satisfied by particular interaction kernels in power-law form, and give a certain characterization of minimizing sets. Finally, following a recent result of Davies, Lim and McCann, we give sufficient conditions on the interaction kernel so that the minimizer of the energy over probability measures is given by Dirac masses concentrated on the vertices of a regular -gon of side length 1 in .
Paper Structure (7 sections, 12 theorems, 89 equations, 2 figures)

This paper contains 7 sections, 12 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Notation and preliminary results
  4. Existence of minimal sets with small volume
  5. Existence results for general kernels
  6. More precise results for power-law kernels
  7. Last results for general kernels

Key Result

Theorem 1.1

Let $N\geq 2$, and $g=g_2$ be given by (defprot). If $\beta=2<\alpha<4$, then the unique minimizer of (eq:problem-with-measures), up to rigid motion, is given by the uniform distribution over a sphere, that is, $\mu = c {\hbox{H}\,\,}^{N-1}\mathop{}\nolimits \partial B_r$ for a suitable choice of $c

Figures (2)

  • Figure 1: The construction in Proposition \ref{['prop:diameter-bound']}.
  • Figure 2: Left: the graph of $g$ must be in the shaded region. Right: the points $x,\,y$ and $z$ and the sets $A_x,\,A_y$ and $Q_{x,y}$ in Step II.

Theorems & Definitions (26)

  • Theorem 1.1: Davies--Lim--McCann, DLM1DLM2
  • Proposition 2.1
  • proof : Sketch of the proof
  • Lemma 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 16 more