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Stability of the ball for attractive-repulsive energies

Marco Bonacini, Riccardo Cristoferi, Ihsan Topaloglu

Abstract

We consider a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. It has recently been proved by R. Frank and E. Lieb that the ball is the unique (up to translation) global minimizer for sufficiently large mass. We focus on the issue of the stability of the ball, in the sense of the positivity of the second variation of the energy with respect to smooth perturbations of the boundary of the ball. We characterize the range of masses for which the second variation is positive definite (large masses) or negative definite (small masses). Moreover, we prove that the stability of the ball implies its local minimality among sets sufficiently close in the Hausdorff distance, but not in $L^1$-sense.

Stability of the ball for attractive-repulsive energies

Abstract

We consider a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. It has recently been proved by R. Frank and E. Lieb that the ball is the unique (up to translation) global minimizer for sufficiently large mass. We focus on the issue of the stability of the ball, in the sense of the positivity of the second variation of the energy with respect to smooth perturbations of the boundary of the ball. We characterize the range of masses for which the second variation is positive definite (large masses) or negative definite (small masses). Moreover, we prove that the stability of the ball implies its local minimality among sets sufficiently close in the Hausdorff distance, but not in -sense.
Paper Structure (9 sections, 12 theorems, 133 equations, 2 figures)

This paper contains 9 sections, 12 theorems, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Stability of the ball
  3. First and second variations
  4. Decomposition in spherical harmonics
  5. The stability thresholds
  6. A Fuglede-type result for nearly-spherical sets
  7. Local optimality of the ball
  8. A necessary condition for L1-local minimality
  9. Explicit values of the stability thresholds

Key Result

Theorem 1.1

Let $\alpha\in(0,d-1)$, $\beta>0$, and let $m_*$, $m_{**}$ be defined by eq:m* and eq:m** respectively. Then: Moreover, for every $m\in(0,m_{**})\cup(m_*,\infty)$ there exist $\overline{C}>0$ and $\bar{\varepsilon}>0$ (depending only on $d$, $\alpha$, $\beta$, and $m$) with the following property: for every measurable set $E\subset\mathbb R^d$ with $|E|=|m|$ and such that for some $x_0\in\mathbb

Figures (2)

  • Figure 1: Numerical plot of the potential $\psi$ of the unit ball, defined in \ref{['eq:potential_ball']}, in the case $d=3$, $\alpha=1$, $\beta=4$ and $\gamma=\frac{1}{7}$. Notice that this value of $\gamma$ is larger than the stability threshold $\gamma_{*}=\frac{1}{8}$ (see Remark \ref{['rmk:L1_min']}) and that the potential does not satisfy the necessary condition in Proposition \ref{['prop:L1-min']}.
  • Figure 2: Numerical plot of the first points of sequence $(X_k)_{k\geqslant2}$, defined in \ref{['app:X_k']}, for two different values of $\beta$, for the choice of parameters $d=3$ and $\alpha=1$ (for which $\beta_*=22$). Left: the case $\beta<\beta_*$ ($\beta=5$). Right: the case $\beta>\beta_*$ ($\beta=155$).

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1: First and second variation of $\mathcal{J}_\sigma$
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6: Properties of $(\mu_k(\sigma))_k$
  • proof
  • ...and 23 more